user7899: In short, I am looking for an online free chess engine that has either:
a.)  multiple engines with multiple ratings, some as low as 1500-1600, and some as high as 2200 or better,
or
b.)  is a single engine with adjustable difficulty ratings from about 1500-1600 and up.
To clarify, someone once recommended that when playing alone against the computer, I should find a difficulty setting that I can beat roughly 2/3 of the time, and another that I can beat only about 1/4 of the time.  Of course I'd rather play OTB (the best practice) but life/job/family commitments don't allow for too much of that these days.  
I can readily beat Microsoft's Chess Titans at all levels, including level 10 (or I should say I can beat it at least 3 out of 4 times). Thus, Chess Titans doesn't fit the bill of being able to beat me somewhat more often.  I realize that most purchasable standard chess software has adjustable ratings, but I don't feel like buying any right now.
Lastly, I am aware that Elo ratings for chess engines are a fuzzy subject -- given that they don't make the same kinds of blunders that real humans do, don't give up, play at different levels in different situations, etc.  I am not demanding a highly specific rating.  I merely would like to vaguely know the general level of my computer opponent while I play.
At any rate, any recommendations for a free online chess engine with adjustable difficulty levels, whose elo ratings are roughly known?

user11838: If you download an UCI like Arena, it is posible to install a lot of free engines and some of them are adjustable in terms of ELO (or the equivalent in computers). I've found this very helpful. In the menu Levels - Limit Playing Strenght to ELO, is posible to tweak the ELO almost by 1 point at a time.
Here you have some of the Engines I have installed, with the respective ELOs:
MadChess   400-2000
Rodent        600-2600
Ufim802      700-2000
Cheng         800-2500
Deuterium 1000-2800
Spike         1100-2500
Rybka        1200-2400
Note: Some of this ELOs are not very accurate, but I think the margin is around 80-100 points. MadChess and Rodent are my everyday engines and I found them pretty consistent.
Hope this help!

user3131: You could use lichess which uses stockfish chess engine.
The difficulty AI levels varies from 1350 to 2400.

user825: http://www.computerchess.org.uk/ccrl/4040 gives you a huge list of engines with their CCRL rating. Please note CCRL rating is not directly related to the human FIDE rating, but it's a reasonable estimate in any case.
I recommend you my engine apps. They're free and very popular. Please visit my site to try my chess apps:
http://www.smallchess.com
You'll find a Stockfish app with adjustable playing strength. You'll also find my own chess engine in the SmallChess app. There is also a Chess Mini app, that plays interesting chess at around FIDE 1800 to 1900. Apart from simply just playing, you'll also receive useful feedback, for example:

It's obvious the great Nimzowitch had been winning most of the game. The graph will tell when and how you blunder.
Now, back to the rating. While the SmallFish app (Stockfish engine integrated) allows adjustable playing strength, the rating reported in the app isn't very accurate. The confidence interval is roughly about +-200 Elo. The rating range given in the SmallChess app is much more accurate, because I'd paired it against human players on the FICS server.
I'm sure you'll learn and improve your chess by playing my apps.

7899: <p>In short, I am looking for an online free chess engine that has either:
a.)  multiple engines with multiple ratings, some as low as 1500-1600, and some as high as 2200 or better,
or
b.)  is a single engine with adjustable difficulty ratings from about 1500-1600 and up.</p>

<p>To clarify, someone once recommended that when playing alone against the computer, I should find a difficulty setting that I can beat roughly 2/3 of the time, and another that I can beat only about 1/4 of the time.  Of course I'd rather play OTB (the best practice) but life/job/family commitments don't allow for too much of that these days.  </p>

<p>I can readily beat Microsoft's Chess Titans at all levels, including level 10 (or I should say I can beat it at least 3 out of 4 times). Thus, Chess Titans doesn't fit the bill of being able to beat me somewhat more often.  I realize that most purchasable standard chess software has adjustable ratings, but I don't feel like buying any right now.</p>

<p>Lastly, I am aware that Elo ratings for chess engines are a fuzzy subject -- given that they don't make the same kinds of blunders that real humans do, don't give up, play at different levels in different situations, etc.  I am not demanding a highly specific rating.  I merely would like to vaguely know the general level of my computer opponent while I play.</p>

<p>At any rate, any recommendations for a free online chess engine with adjustable difficulty levels, whose elo ratings are roughly known?</p>
 11838: <p>If you download an UCI like Arena, it is posible to install a lot of free engines and some of them are adjustable in terms of ELO (or the equivalent in computers). I've found this very helpful. In the menu Levels - Limit Playing Strenght to ELO, is posible to tweak the ELO almost by 1 point at a time.</p>

<p>Here you have some of the Engines I have installed, with the respective ELOs:
MadChess   400-2000
Rodent        600-2600
Ufim802      700-2000
Cheng         800-2500
Deuterium 1000-2800
Spike         1100-2500
Rybka        1200-2400</p>

<p>Note: Some of this ELOs are not very accurate, but I think the margin is around 80-100 points. MadChess and Rodent are my everyday engines and I found them pretty consistent.</p>

<p>Hope this help!</p>
 3131: <p>You could use <a href="http://en.lichess.org/" rel="nofollow">lichess</a> which uses stockfish chess engine.</p>

<p>The difficulty AI levels varies from 1350 to 2400.</p>
 825: <p><a href="http://www.computerchess.org.uk/ccrl/4040" rel="nofollow noreferrer">http://www.computerchess.org.uk/ccrl/4040</a> gives you a huge list of engines with their CCRL rating. Please note CCRL rating is not directly related to the human FIDE rating, but it's a reasonable estimate in any case.</p>

<p>I recommend you my engine apps. They're free and very popular. Please visit my site to try my chess apps:</p>

<p><a href="http://http:://www.smallchess.com" rel="nofollow noreferrer">http://www.smallchess.com</a></p>

<p>You'll find a Stockfish app with adjustable playing strength. You'll also find my own chess engine in the SmallChess app. There is also a Chess Mini app, that plays interesting chess at around FIDE 1800 to 1900. Apart from simply just playing, you'll also receive useful feedback, for example:</p>

<p><a href="https://i.stack.imgur.com/F917O.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/F917O.jpg" alt="enter image description here"></a></p>

<p>It's obvious the great Nimzowitch had been winning most of the game. The graph will tell when and how you blunder.</p>

<p>Now, back to the rating. While the SmallFish app (Stockfish engine integrated) allows adjustable playing strength, the rating reported in the app isn't very accurate. The confidence interval is roughly about +-200 Elo. The rating range given in the SmallChess app is much more accurate, because I'd paired it against human players on the FICS server.</p>

<p>I'm sure you'll learn and improve your chess by playing my apps.</p>
