user264: I'm looking for a Chess tutoring/teaching app that I can dip in and out of on my phone. At the moment I use Shredder Chess to practice when on the move (which basically involves repeating the puzzles over and over) but I was looking for some sort of app which has structured lessons on openings, defenses etc. and explains the positions in depth like Chess Mentor does. 
Does anyone know if this type of app exists? 

user3456: Rather than installing an app you can use the mobile interface of the tactics trainer at http://chesstempo.com.

user4545: For opening training using GM stats and engine analysis: http://www.chessopeningsapp.com

user3890: For the several facets of the game, you may use different apps. I include them all on the comments, since I'm not allowed to post them here.
Tactics: This very important part of the game can be trained by the already mentioned app "Tactics Trainer". I'm a daily user, and it's excellent.
Openings: The best app I can think of in this area is "Chess Repertoire". The app is excellent, with a nice clean design and with more opening positions that you'll need. It even has space for personal annotations and novelties. The sole drawback is that only the first moves are free, and you've to pay for the rest of it.
Endgame: I've been recommended "Chess Endings", which is not free, and cannot say much about it. You may try it.
Casual play: I strongly suggest the Chess.com app. You have a few additional tactics problems, but the interesting thing is the correspondence and live chess, playing whenever you feel.
Analysis: With abysmal difference, the best analysis tool you can possess for a mobile device is "Scid". It is beautiful, the pieces are shown clearly, there's space for the game score and admits an engine. It also allows you to export games to .pgn and many more things. Priceless, but open source and free.
Despite it comes with an engine by default, Stockfish, downloadable from here, can be compiled into any Linux device, and it's certainly the best option if you possess an Android.

user4389: Try Chess Run
It's a chess-based platform game - not a pure chess game but a platform game featuring the standard chess pieces which are used to cross a burning board filled with enemy pieces.
You have the option of highlighting squares that your pieces can move to so it's a good way of learning the moves. It's also fast-paced so will help with speed chess.

user10037: https://itunes.apple.com/ca/app/chesstutorapp/id1096289047
ChessTutor - rates your moves as you play through Master games.
PS: This is my app.

user1657: For Android you can also try Chesster. It lets you import an ebook with chess moves, and makes it interactive, showing the movements in a chessboard. 

user2555: You can try Christal Kernel's Chess Wizard for iOS.
It trains you while playing. It gives a feeling that there is a real coach around. Throws situation based puzzle on the board. Makes the game interesting for the learners.
Detailed information on their website.

user7648: For Android check out Chess Endgames.  It's free and has a number of different positions to work through.

user7507: I know there are already a lot of answers here, but one that isn't mentioned is the Chess24 app. I really like the tactics trainer from the app and find that for the money it is a better value than others I've seen. (Should still be $20 for 1yr of unlimited tactics.) There are also great apps for iOS at least that allow you to read chess books and follow along with a chessboard like Forward Chess or e+books Chess. I have found this to be incredibly helpful in studying.

user1262: Try this app, Tactics Trainer. It has helped me a lot and has a lot of puzzles, but I dunno if it has an iOS version.
I play at chess.com, which has an app, too, for android and iOs. I play with real people and it has 3 free puzzles per day.
Both apps have some graphics about your games or puzzles, and your ELO rating.
Edit:
By the way, chessimo is really good but I don't think it has a free version, or it is just a demo; to get the full version you have to pay, but it is really good, it was the first Tactics trainer app I used.

user8782: I recommend Chess King Learn for chess education.
The apps are developed by our team and can help chess lovers of any level and age to study chess.
The app is presented for Android, iOS and macOS platforms. Also there is a web version. You can study on any platform you want.
You may choose your course in the apps linked above, they include 90+ chess courses. Each course has a free part, so there are thousands of free puzzles in total. If you'll fall in love with one of the courses, you may purchase the full version ;).
Beside the main app there are more than 60 single applications where you can focus on a specific course. Here is the list of top rated ones:

CT-ART 4.0 (Chess Tactics 1200-2400 ELO)
Chess: From Beginners To Club
Chess Tactics for Beginners
Total Chess Ending (1600 - 2400)
Chess Opening Blunders
Chess Guide Vol. 1 (1600 - 2000)
Chess Guide Vol. 2 (1800 - 2200)
Chess Strategy (1800 - 2400)
Chess Tactics Art (1400 - 1600)
Chess Tactics Art (1600 - 1800)
Chess Tactics Art: Mating
Chess Combinations Vol. 1
Chess Combinations Vol. 2
Chess Puzzles Advanced Defense

Full apps list is here.

user1197: Try the Chess.com app.
It allows you to play against real people and against the computer. It has puzzles based on tactics to solve & videos by experts.
Android Version (requires Android OS 1.5 or later)
iPhone Version (requires iOS 3.0 or later)

user1193: The one that I could not recommend more for iOS is Chessimo.
This has great ability to train you for tactics, strategy and endgame. I teach a few people how to play chess well and this method seems to be very helpful. If you want to get to a decent level fast- focus on tactics, then strategy, then endgame and at the end opening.
This program makes it easy and fun!

user1275: Here are my Chess apps for Android:
iChess — lots of chess tactics to solve, all offline. You can also load your own PGN or purchase recent tactics based on openings, or from recent games.
Chess Book Study — a very innovative app that shows a board and an eBook in the same screen. It makes it easy to study chess books. You can even dictate and the moves are made on the board.
Analyze This — analyze a position with two engines, simultaneously!

264: <p>I'm looking for a Chess tutoring/teaching app that I can dip in and out of on my phone. At the moment I use <a href="https://play.google.com/store/apps/details?id=com.shredderchess.android&amp;hl=en" rel="noreferrer">Shredder Chess</a> to practice when on the move (which basically involves repeating the puzzles over and over) but I was looking for some sort of app which has structured lessons on openings, defenses etc. and explains the positions in depth like <a href="https://www.chess.com/chessmentor/" rel="noreferrer">Chess Mentor</a> does. </p>

<p>Does anyone know if this type of app exists? </p>
 3456: <p>Rather than installing an app you can use the mobile interface of the tactics trainer at <a href="http://chesstempo.com" rel="nofollow">http://chesstempo.com</a>.</p>
 4545: <p>For opening training using GM stats and engine analysis: <a href="http://www.chessopeningsapp.com" rel="nofollow">http://www.chessopeningsapp.com</a></p>
 3890: <p>For the several facets of the game, you may use different apps. I include them all on the comments, since I'm not allowed to post them here.</p>

<p><strong>Tactics:</strong> This very important part of the game can be trained by the already mentioned app "Tactics Trainer". I'm a daily user, and it's excellent.</p>

<p><strong>Openings:</strong> The best app I can think of in this area is "Chess Repertoire". The app is excellent, with a nice clean design and with more opening positions that you'll need. It even has space for personal annotations and novelties. The sole drawback is that only the first moves are free, and you've to pay for the rest of it.</p>

<p><strong>Endgame:</strong> I've been recommended "Chess Endings", which is not free, and cannot say much about it. You may try it.</p>

<p><strong>Casual play:</strong> I strongly suggest the <a href="http://www.chess.com" rel="nofollow noreferrer">Chess.com</a> app. You have a few additional tactics problems, but the interesting thing is the correspondence and live chess, playing whenever you feel.</p>

<p><strong>Analysis:</strong> With abysmal difference, the best analysis tool you can possess for a mobile device is "Scid". It is beautiful, the pieces are shown clearly, there's space for the game score and admits an engine. It also allows you to export games to .pgn and many more things. Priceless, but open source and free.
Despite it comes with an engine by default, Stockfish, downloadable from <a href="http://stockfishchess.org/" rel="nofollow noreferrer">here</a>, can be compiled into any Linux device, and it's certainly the best option if you possess an Android.</p>
 4389: <p>Try <a href="https://play.google.com/store/apps/details?id=com.dorisstreetgames.chessrun" rel="nofollow" title="Chess Run">Chess Run</a></p>

<p>It's a chess-based platform game - not a pure chess game but a platform game featuring the standard chess pieces which are used to cross a burning board filled with enemy pieces.
You have the option of highlighting squares that your pieces can move to so it's a good way of learning the moves. It's also fast-paced so will help with speed chess.</p>
 10037: <p><a href="https://itunes.apple.com/ca/app/chesstutorapp/id1096289047" rel="nofollow">https://itunes.apple.com/ca/app/chesstutorapp/id1096289047</a></p>

<p>ChessTutor - rates your moves as you play through Master games.</p>

<p>PS: This is my app.</p>
 1657: <p>For Android you can also try <a href="https://play.google.com/store/apps/details?id=com.bento.chessterlite" rel="nofollow">Chesster</a>. It lets you import an ebook with chess moves, and makes it interactive, showing the movements in a chessboard. </p>
 2555: <p>You can try Christal Kernel's <a href="https://itunes.apple.com/in/app/chess-wizard/id916537412?mt=8" rel="nofollow">Chess Wizard</a> for iOS.</p>

<p>It trains you while playing. It gives a feeling that there is a real coach around. Throws situation based puzzle on the board. Makes the game interesting for the learners.</p>

<p><a href="http://crystalkernel.com" rel="nofollow">Detailed information on their website</a>.</p>
 7648: <p>For Android check out <a href="https://play.google.com/store/apps/details?id=com.completethink.chessendgames" rel="nofollow">Chess Endgames</a>.  It's free and has a number of different positions to work through.</p>
 7507: <p>I know there are already a lot of answers here, but one that isn't mentioned is the Chess24 app. I really like the tactics trainer from the app and find that for the money it is a better value than others I've seen. (Should still be $20 for 1yr of unlimited tactics.) There are also great apps for iOS at least that allow you to read chess books and follow along with a chessboard like Forward Chess or e+books Chess. I have found this to be incredibly helpful in studying.</p>
 1262: <p>Try this app, <a href="https://play.google.com/store/apps/details?id=com.thecvsi.tactictrainer&amp;hl=es" rel="nofollow">Tactics Trainer</a>. It has helped me a lot and has a lot of puzzles, but I dunno if it has an iOS version.</p>

<p>I play at <a href="http://www.chess.com" rel="nofollow">chess.com</a>, which has an app, too, for android and iOs. I play with real people and it has 3 free puzzles per day.</p>

<p>Both apps have some graphics about your games or puzzles, and your ELO rating.</p>

<p>Edit:</p>

<p>By the way, <a href="http://www.chessimo.com" rel="nofollow">chessimo</a> is really good but I don't think it has a free version, or it is just a demo; to get the full version you have to pay, but it is really good, it was the first Tactics trainer app I used.</p>
 8782: <p>I recommend <a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn" rel="nofollow noreferrer">Chess King Learn</a> for chess education.
The apps are developed by our team and can help chess lovers of any level and age to study chess.</p>
<p>The app is presented for <a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn" rel="nofollow noreferrer">Android</a>, <a href="https://apps.apple.com/us/app/id1305823239" rel="nofollow noreferrer">iOS</a> and <a href="https://apps.apple.com/us/app/id1305823239" rel="nofollow noreferrer">macOS</a> platforms. Also there is a <a href="https://learn.chessking.com/" rel="nofollow noreferrer">web</a> version. You can study on any platform you want.</p>
<p>You may choose your course in the apps linked above, they include 90+ chess courses. Each course has a free part, so there are thousands of free puzzles in total. If you'll fall in love with one of the courses, you may purchase the full version ;).</p>
<p>Beside the main app there are more than 60 single applications where you can focus on a specific course. Here is the list of top rated ones:</p>
<ul>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.ctart4" rel="nofollow noreferrer">CT-ART 4.0 (Chess Tactics 1200-2400 ELO)</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.beginnerstoclub" rel="nofollow noreferrer">Chess: From Beginners To Club</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.ctforbeginners" rel="nofollow noreferrer">Chess Tactics for Beginners</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.totalchessending" rel="nofollow noreferrer">Total Chess Ending (1600 - 2400)</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.openingblunders" rel="nofollow noreferrer">Chess Opening Blunders</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.guideforclub" rel="nofollow noreferrer">Chess Guide Vol. 1 (1600 - 2000)</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.guideforintermediates" rel="nofollow noreferrer">Chess Guide Vol. 2 (1800 - 2200)</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.chessstrategy" rel="nofollow noreferrer">Chess Strategy (1800 - 2400)</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.ctartforbeginners" rel="nofollow noreferrer">Chess Tactics Art (1400 - 1600)</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.ctartforclub" rel="nofollow noreferrer">Chess Tactics Art (1600 - 1800)</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.ctartmating" rel="nofollow noreferrer">Chess Tactics Art: Mating</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.chesscombinations1" rel="nofollow noreferrer">Chess Combinations Vol. 1</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.chesscombinations2" rel="nofollow noreferrer">Chess Combinations Vol. 2</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.chessking.android.learn.advanceddefense" rel="nofollow noreferrer">Chess Puzzles Advanced Defense</a></li>
</ul>
<p>Full apps list is <a href="https://play.google.com/store/apps/dev?id=8196632901699712832" rel="nofollow noreferrer">here</a>.</p>
 1197: <p>Try the Chess.com app.<br />
It allows you to play against real people and against the computer. It has puzzles based on tactics to solve &amp; videos by experts.</p>

<p><a href="http://www.chess.com/play/android" rel="nofollow">Android Version</a> (requires Android OS 1.5 or later)</p>

<p><a href="http://www.chess.com/play/iphone" rel="nofollow">iPhone Version</a> (requires iOS 3.0 or later)</p>
 1193: <p>The one that I could not recommend more for iOS is <a href="https://itunes.apple.com/gb/app/chessimo/id504839878?mt=8" rel="nofollow">Chessimo</a>.</p>

<p>This has great ability to train you for tactics, strategy and endgame. I teach a few people how to play chess well and this method seems to be very helpful. If you want to get to a decent level fast- focus on tactics, then strategy, then endgame and at the end opening.</p>

<p>This program makes it easy and fun!</p>
 1275: <p>Here are <a href="https://play.google.com/store/apps/collection/cluster?clp=igM4ChkKEzU5ODkyOTc1ODg2OTk4MjEzMDQQCBgDEhkKEzU5ODkyOTc1ODg2OTk4MjEzMDQQCBgDGAA%3D:S:ANO1ljJGNNw&amp;gsr=CjuKAzgKGQoTNTk4OTI5NzU4ODY5OTgyMTMwNBAIGAMSGQoTNTk4OTI5NzU4ODY5OTgyMTMwNBAIGAMYAA%3D%3D:S:ANO1ljKOLmE" rel="nofollow noreferrer">my Chess apps for Android</a>:</p>

<p><strong>iChess</strong> — lots of chess tactics to solve, all offline. You can also load your own PGN or purchase recent tactics based on openings, or from recent games.</p>

<p><strong>Chess Book Study</strong> — a very innovative app that shows a board and an eBook in the same screen. It makes it easy to study chess books. You can even dictate and the moves are made on the board.</p>

<p><strong>Analyze This</strong> — analyze a position with two engines, simultaneously!</p>
