user259: I would like to learn about a few checkmates using very few moves. I know one where you develop the knight and march it directly towards the King. Does anyone know the name of this mate and suggest others?

user8201: Here’s a simple example which I always like.
[FEN ""]
1. d4 e5 2. Bg5 Qxg5 3. Qc1 Qxc1#



user924: Against the Bird Opening (1.f4), black can play the From Gambit which threatens a kind of fool's mate:
[FEN ""]

1.f4 e5 2.fxe5 d6 3.exd6 Bxd6 4.Nc3? Qh4+ 5.g3 Qxg3+ 6.hxg3 Bxg3#

Instead of 4.Nc3, which is just an example move, 4.d4 and 4.g3 are common.

user10380: Légal's mate, which Andrew mentioned, can also occur in the Italian Opening, with a move order which is perhaps more credible:
1.e4 e5 2. Nf3 Nc6 3. Bc4 d6 4. Nc3 Bg4? 5. h3 Bh5? 6. Nxe5! Bxd1? 7. Bxf7+! Ke7 8. Nd5#.
Black has a similar trap in the Petrov:
1.e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6? 4. Nxc6 dxc6 5. d3 Bc5 6. Bg5? Nxe4! 7. Bxd8? Bxf2+ 8. Ke2 Bg4#.
John Coleman mentioned the Kieninger Trap, a Black trap in in the Budapest gambit. White has two traps in the Caro-Kann that are similar to this:
1.e4 c6 2. d4 d5 3. Nc3 dxe4 4. Nxe4 Nd7 {Nf6 is better} 5. Qe2?! Ngf6? 5. Nd6#
This also works after 2. Nc3 d5 3. Nf3.

user15148: The Kieninger Trap in the Budapest gambit has trapped numerous amateurs but even some masters over the years. It involves a similar mating pattern to those found in some of the other answers. There are only so many ways to blunder a king early in the game.
1.d4 Nf6
2.c4 e5
3.dxe5 Ng4
4.Bf4 Nc6
5.Nf3 Bb4+
6.Nbd2 Qe7
7.a3 Ngxe5
8.axb4 Nd3#
[FEN ""]
1.d4 Nf6 
2.c4 e5 
3.dxe5 Ng4 
4.Bf4 Nc6 
5.Nf3 Bb4+ 
6.Nbd2 Qe7 
7.a3 Ngxe5 
8.axb4 Nd3# 


user3456: [FEN ""]

1. d4 Nf6 2. c4 c5 3. d5 b5 4. cxb5 a6 5. Nc3 axb5 6. e4 b4 7. Nb5 Nxe4 8. Qe2 Nf6 9. Nd6#


user15223: To add to the Fool's Mate idea, this game left enough of an impression on me to remember it.
[FEN ""]
1. d4 f5 2. Bg5 {An enterprising line, threatening to weaken Black's pawn structure after the natural Nf6.} h6 3. Bh4 g5!? {This seems to trap the bishop, but the holes around the Black king are very dangerous. It is objectively OK, but only if Black does not play to win material.} 4. Bg3 f4? {This move is a definite mistake.} 5. e3! {Now fxg3 leads to Qh5#, and White threatens to rescue his bishop with exf4} h5 {Renewing the threat, now 6. exf4 h4 and the bishop is still trapped} 6. Bd3 {Objectively this move isn't the best, but it threatens Bg6#} Rh6?? 7. Qxh5+! Rxh5 8. Bg6#


user227: There's a 2-move checkmate, termed fool's mate:


f3 e5 2. g4 Qh4#


Fool's mate on Wikipedia

user8393: [FEN "nbqnbrkr/pppppppp/8/8/8/8/PPPPPPPP/NBQNBRKR w KQkq - 0 1"]
[StartPly "3"]

1. d3 {Pyt Honisten eröffnet mit seinem Lieblingsbauern} g5 {Retho Riker antwortet mit seiner Standard-Antwort} 2. Qxg5# {Pyt Honisten ging zum nächsten Brett, während sich Retho Riker tränenreich vom Schach verabschiedete.}

With Chess960, also known as Fisher Random Chess, you can achieve it, too.

user12629: The Blackburne-Shilling Gambit could result in a quick loss for White if he plays some fairly plausible moves:
1.e4 e5 2.Nf3 Nc6 3.Bc4 {Italian opening} Nd4 4.Nxe5 {Free pawn!} Qg5 5.Nxf7 {Forking the queen and rook.} Qxg2 6.Rf1 {Preventing Qxh1} Qxe4+ 7.Be2 {Qe2 Nxe2 drops the queen} Nf3#
[FEN ""]
1.e4 e5 2.Nf3 Nc6 3.Bc4 {Italian opening} Nd4 4.Nxe5 {Free pawn!} Qg5 5.Nxf7 {Forking the queen and rook.} Qxg2 6.Rf1 {Preventing Qxh1} Qxe4+ 7.Be2 {Qe2 Nxe2 drops the queen} Nf3#  0-1


user2041: A mate in 2½ move, derived from Wayward Queen Attack, occurs if black moves Ke7 as a panic attempt to "protect" the e5 pawn with the king, which may occur if young players move too quickly:
[FEN ""]

1. e4 e5 
2. Qh5 Ke7 
3. Qe5#


user-1: The Scholar's and Fool's mates that others have cited are the shortest.  They rely on two different but highly important principles chess players should remember.  Scholar's mate relies on the opponent's insufficiently protected f2 (or ...f7) point, while Fool's mate relies on a weakening of the diagonal from h5 (or ...h4) to the opponent's king.
While you can actively try for Scholar's mate (and many weak amateurs fall for it), Fool's mate pretty much relies on very poor play by the opponent -- advancing the f and g pawns such that the critical diagonal is fatally weakened.  A variant of Fool's mate is more likely to occur that involves several more moves and induces the opponent to make the required weakening moves.  Consider the following sequence; White's fourth move is a fatal mistake leading to the loss of his queen or getting mated along the ...h4-e1 diagonal.
[FEN ""]
1. d4 Nf6 2. Nd2 e5 3. dxe5 Ng4 4. h3?? Ne3 5. fxe3 Qh4+ 6. g3 Qxg3#


An excellent old book that should probably be a new player's second or third book is "The Art of Checkmate", despite being so old it still uses descriptive notation:
http://www.amazon.com/The-Art-Checkmate-Dover-Chess/dp/0486201066

user167: For a short mating possibility in a very standard mainline opening, one can look to the Smyslov-Karpov variation in the Caro-Kann Defense:
[FEN ""]
[Result "1-0"]
1.e4 c6 2.d4 d5 3.Nc3 dxe4 4.Nxe4 Nd7 5.Qe2 Ngf6?? 6.Nd6#

The most popular continuations are 5. Nf3, 5. Bc4 and 5. Ng5 (an aggressive try which Kasparov assayed a number of times against Karpov). The sneaky try 5. Qe2 just sets up a pin against Black's e-pawn, making the intended followup 5... Ngf6?? unplayable.
There are a mere 271 games in my database (out of several million) in which this 5. Qe2 appears, but one finds a few GMs giving it a go: Ivan Radulov, Oleg Nikolenko, Joe Gallagher (against as strong an opponent as IM Tibor Karolyi), even some guy named Alekhine (though, admittedly, only in a consultation game against a team of four amateurs). A 2300 player also tried it out against Yasser Seirawan on the black side; Seirawan didn't fall into the trap of course, but the 2300 did manage to get a draw out of the game.
If you're still reading, you're probably curious: out of the 271 games I have on record, in only 34 of them did black fall into the trap with 5... Ngf6. Alekhine's amateur foes were among the victims, and the highest rated victim was in a game Hans Speck (2140) - Nikolaus Mertens (2165) from the 1997 Liechtenstein Open. Most interestingly, on two of those 34 occasions, the white player failed to reply with 6. Nd6#. One was a player rated 1873 who went on to lose to his 2202 opponent; the other was rated 2264(!) and managed to win anyway against a 2113 on the black side.

user12: There are probably as many quick mates as there have been first time players (since we all get caught in these at the start!) but here are a few common ones:
Fool's Mate
This is the shortest possible checkmate in a mere two moves:
1.f4 e5 2. g4 Qh4#
[FEN ""]

1. f4 e5 2. g4 Qh4# 0-1


White can also play f3 instead of f4 or move the g pawn before the f pawn.
Scholar's Mate
This mating pattern is also sometimes referred to as "Fool's Mate" and there are two main patterns:
1.e4 e5 2. Bc4 (2. Qh5 Nc6 3. Bc4 Nf6 4. Qxf7#) Nc6 3. Qf3 Bc5 4. Qxf7#
[FEN ""]

1. e4 e5 2. Bc4 (2. Qh5 Nc6 3. Bc4 Nf6 4. Qxf7#) Nc6 3. Qf3 Bc5 4. Qxf7# 1-0


The key is that f7 is insufficiently defended and mate is delivered there.
Légal's Mate
Légal's Mate (also spelled as Légall's Mate) is probably the mate referenced in the question:
1.e4 e5 2.Nf3 Nc6 3.Bc4 d6 4.Nc3 Bg4  5.h3 Bh5? 6.Nxe5! Bxd1?? 7.Bxf7+ Ke7 8.Nd5#
[FEN ""]

1.e4 e5 2.Nf3 Nc6 3.Bc4 d6 4.Nc3 Bg4  
5.h3 Bh5? 6.Nxe5! Bxd1?? 7.Bxf7+ Ke7 8.Nd5# 1-0


259: <p>I would like to learn about a few checkmates using very few moves. I know one where you develop the knight and march it directly towards the King. Does anyone know the name of this mate and suggest others?</p>
 8201: <p>Here’s a simple example which I always like.</p>
<pre><code>[FEN &quot;&quot;]
1. d4 e5 2. Bg5 Qxg5 3. Qc1 Qxc1#
</code></pre>
<br/>
 924: <p>Against the Bird Opening (1.f4), black can play the From Gambit which threatens a kind of fool's mate:</p>
<pre><code>[FEN &quot;&quot;]

1.f4 e5 2.fxe5 d6 3.exd6 Bxd6 4.Nc3? Qh4+ 5.g3 Qxg3+ 6.hxg3 Bxg3#
</code></pre>
<p>Instead of 4.Nc3, which is just an example move, 4.d4 and 4.g3 are common.</p>
 10380: <p>Légal's mate, which <a href="https://chess.stackexchange.com/a/585">Andrew mentioned</a>, can also occur in the Italian Opening, with a move order which is perhaps more credible:</p>
<p>1.e4 e5 2. Nf3 Nc6 3. Bc4 d6 4. Nc3 Bg4? 5. h3 Bh5? 6. Nxe5! Bxd1? 7. Bxf7+! Ke7 8. Nd5#.</p>
<p>Black has a similar trap in the Petrov:</p>
<p>1.e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6? 4. Nxc6 dxc6 5. d3 Bc5 6. Bg5? Nxe4! 7. Bxd8? Bxf2+ 8. Ke2 Bg4#.</p>
<p><a href="https://chess.stackexchange.com/a/34308">John Coleman mentioned</a> the Kieninger Trap, a Black trap in in the Budapest gambit. White has two traps in the Caro-Kann that are similar to this:</p>
<p>1.e4 c6 2. d4 d5 3. Nc3 dxe4 4. Nxe4 Nd7 {Nf6 is better} 5. Qe2?! Ngf6? 5. Nd6#</p>
<p>This also works after 2. Nc3 d5 3. Nf3.</p>
 15148: <p>The <a href="https://en.wikipedia.org/wiki/Budapest_Gambit#Kieninger_Trap" rel="nofollow noreferrer">Kieninger Trap</a> in the Budapest gambit has trapped numerous amateurs but even some masters over the years. It involves a similar mating pattern to those found in some of the other answers. There are only so many ways to blunder a king early in the game.</p>
<p>1.d4 Nf6
2.c4 e5
3.dxe5 Ng4
4.Bf4 Nc6
5.Nf3 Bb4+
6.Nbd2 Qe7
7.a3 Ngxe5
8.axb4 Nd3#</p>
<pre><code>[FEN &quot;&quot;]
1.d4 Nf6 
2.c4 e5 
3.dxe5 Ng4 
4.Bf4 Nc6 
5.Nf3 Bb4+ 
6.Nbd2 Qe7 
7.a3 Ngxe5 
8.axb4 Nd3# 
</code></pre>
 3456: <pre><code>[FEN ""]

1. d4 Nf6 2. c4 c5 3. d5 b5 4. cxb5 a6 5. Nc3 axb5 6. e4 b4 7. Nb5 Nxe4 8. Qe2 Nf6 9. Nd6#
</code></pre>
 15223: <p>To add to the Fool's Mate idea, this game left enough of an impression on me to remember it.</p>

<pre><code>[FEN ""]
1. d4 f5 2. Bg5 {An enterprising line, threatening to weaken Black's pawn structure after the natural Nf6.} h6 3. Bh4 g5!? {This seems to trap the bishop, but the holes around the Black king are very dangerous. It is objectively OK, but only if Black does not play to win material.} 4. Bg3 f4? {This move is a definite mistake.} 5. e3! {Now fxg3 leads to Qh5#, and White threatens to rescue his bishop with exf4} h5 {Renewing the threat, now 6. exf4 h4 and the bishop is still trapped} 6. Bd3 {Objectively this move isn't the best, but it threatens Bg6#} Rh6?? 7. Qxh5+! Rxh5 8. Bg6#
</code></pre>
 227: <p>There's a 2-move checkmate, termed <a href="http://www.youtube.com/watch?v=vFKxKviVL1I" rel="nofollow noreferrer">fool's mate</a>:</p>

<blockquote>
  <ol>
  <li>f3 e5 2. g4 Qh4#</li>
  </ol>
</blockquote>

<p><a href="http://en.wikipedia.org/wiki/Fool%27s_mate" rel="nofollow noreferrer">Fool's mate on Wikipedia</a></p>
 8393: <pre><code>[FEN "nbqnbrkr/pppppppp/8/8/8/8/PPPPPPPP/NBQNBRKR w KQkq - 0 1"]
[StartPly "3"]

1. d3 {Pyt Honisten eröffnet mit seinem Lieblingsbauern} g5 {Retho Riker antwortet mit seiner Standard-Antwort} 2. Qxg5# {Pyt Honisten ging zum nächsten Brett, während sich Retho Riker tränenreich vom Schach verabschiedete.}
</code></pre>

<p>With Chess960, also known as Fisher Random Chess, you can achieve it, too.</p>
 12629: <p>The Blackburne-Shilling Gambit could result in a quick loss for White if he plays some fairly plausible moves:</p>
<p>1.e4 e5 2.Nf3 Nc6 3.Bc4 {Italian opening} Nd4 4.Nxe5 {Free pawn!} Qg5 5.Nxf7 {Forking the queen and rook.} Qxg2 6.Rf1 {Preventing Qxh1} Qxe4+ 7.Be2 {Qe2 Nxe2 drops the queen} Nf3#</p>
<pre><code>[FEN &quot;&quot;]
1.e4 e5 2.Nf3 Nc6 3.Bc4 {Italian opening} Nd4 4.Nxe5 {Free pawn!} Qg5 5.Nxf7 {Forking the queen and rook.} Qxg2 6.Rf1 {Preventing Qxh1} Qxe4+ 7.Be2 {Qe2 Nxe2 drops the queen} Nf3#  0-1
</code></pre>
 2041: <p>A mate in 2½ move, derived from <a href="https://en.wikipedia.org/wiki/Wayward_Queen_Attack" rel="noreferrer">Wayward Queen Attack</a>, occurs if black moves Ke7 as a panic attempt to "protect" the e5 pawn with the king, which may occur if young players move too quickly:</p>

<pre><code>[FEN ""]

1. e4 e5 
2. Qh5 Ke7 
3. Qe5#
</code></pre>
 -1: <p>The Scholar's and Fool's mates that others have cited are the shortest.  They rely on two different but highly important principles chess players should remember.  Scholar's mate relies on the opponent's insufficiently protected f2 (or ...f7) point, while Fool's mate relies on a weakening of the diagonal from h5 (or ...h4) to the opponent's king.</p>

<p>While you can actively try for Scholar's mate (and many weak amateurs fall for it), Fool's mate pretty much relies on very poor play by the opponent -- advancing the f and g pawns such that the critical diagonal is fatally weakened.  A variant of Fool's mate is more likely to occur that involves several more moves and induces the opponent to make the required weakening moves.  Consider the following sequence; White's fourth move is a fatal mistake leading to the loss of his queen or getting mated along the ...h4-e1 diagonal.</p>

<pre><code>[FEN ""]
1. d4 Nf6 2. Nd2 e5 3. dxe5 Ng4 4. h3?? Ne3 5. fxe3 Qh4+ 6. g3 Qxg3#
</code></pre>

<p><br />
An excellent old book that should probably be a new player's second or third book is "The Art of Checkmate", despite being so old it still uses descriptive notation:</p>

<p><a href="http://rads.stackoverflow.com/amzn/click/0486201066">http://www.amazon.com/The-Art-Checkmate-Dover-Chess/dp/0486201066</a></p>
 167: <p>For a short mating possibility in a very standard mainline opening, one can look to the <a href="http://en.wikipedia.org/wiki/Caro-Kann_Defence#Smyslov_.2F_Karpov_.2F_Modern_Variation">Smyslov-Karpov variation</a> in the Caro-Kann Defense:</p>

<pre><code>[FEN ""]
[Result "1-0"]
1.e4 c6 2.d4 d5 3.Nc3 dxe4 4.Nxe4 Nd7 5.Qe2 Ngf6?? 6.Nd6#
</code></pre>

<p>The most popular continuations are <code>5. Nf3</code>, <code>5. Bc4</code> and <code>5. Ng5</code> (an aggressive try which Kasparov assayed a number of times against Karpov). The sneaky try <code>5. Qe2</code> just sets up a pin against Black's e-pawn, making the intended followup <code>5... Ngf6??</code> unplayable.</p>

<p>There are a mere 271 games in my database (out of several million) in which this 5. Qe2 appears, but one finds a few GMs giving it a go: <a href="http://en.wikipedia.org/wiki/Ivan_Radulov">Ivan Radulov</a>, <a href="http://www.chessgames.com/player/oleg_nikolenko.html">Oleg Nikolenko</a>, <a href="http://en.wikipedia.org/wiki/Joseph_Gallagher">Joe Gallagher</a> (against as strong an opponent as IM <a href="http://en.wikipedia.org/wiki/Tibor_K%C3%A1rolyi_%28chess_player%29">Tibor Karolyi</a>), even some guy named <a href="http://en.wikipedia.org/wiki/Alexander_Alekhine">Alekhine</a> (though, admittedly, only in a consultation game against a team of four amateurs). A 2300 player also tried it out against <a href="http://en.wikipedia.org/wiki/Yasser_Seirawan">Yasser Seirawan</a> on the black side; Seirawan didn't fall into the trap of course, but the 2300 did manage to get a draw out of the game.</p>

<p>If you're still reading, you're probably curious: out of the 271 games I have on record, in only 34 of them did black fall into the trap with <code>5... Ngf6</code>. Alekhine's amateur foes were among the victims, and the highest rated victim was in a game Hans Speck (2140) - Nikolaus Mertens (2165) from the 1997 Liechtenstein Open. <strong>Most interestingly</strong>, on two of those 34 occasions, the white player failed to reply with <code>6. Nd6#</code>. One was a player rated 1873 who went on to lose to his 2202 opponent; the other was rated 2264(!) and managed to win anyway against a 2113 on the black side.</p>
 12: <p>There are probably as many quick mates as there have been first time players (since we all get caught in these at the start!) but here are a few common ones:</p>
<h2>Fool's Mate</h2>
<p>This is the shortest possible checkmate in a mere two moves:</p>
<p>1.f4 e5 2. g4 Qh4#</p>
<pre><code>[FEN &quot;&quot;]

1. f4 e5 2. g4 Qh4# 0-1
</code></pre>
<br />
White can also play f3 instead of f4 or move the g pawn before the f pawn.
<h2>Scholar's Mate</h2>
<p>This mating pattern is also sometimes referred to as &quot;Fool's Mate&quot; and there are two main patterns:</p>
<p>1.e4 e5 2. Bc4 (2. Qh5 Nc6 3. Bc4 Nf6 4. Qxf7#) Nc6 3. Qf3 Bc5 4. Qxf7#</p>
<pre><code>[FEN &quot;&quot;]

1. e4 e5 2. Bc4 (2. Qh5 Nc6 3. Bc4 Nf6 4. Qxf7#) Nc6 3. Qf3 Bc5 4. Qxf7# 1-0
</code></pre>
<br />
The key is that f7 is insufficiently defended and mate is delivered there.
<h2>Légal's Mate</h2>
<p>Légal's Mate (also spelled as Légall's Mate) is probably the mate referenced in the question:</p>
<p>1.e4 e5 2.Nf3 Nc6 3.Bc4 d6 4.Nc3 Bg4  5.h3 Bh5? 6.Nxe5! Bxd1?? 7.Bxf7+ Ke7 8.Nd5#</p>
<pre><code>[FEN &quot;&quot;]

1.e4 e5 2.Nf3 Nc6 3.Bc4 d6 4.Nc3 Bg4  
5.h3 Bh5? 6.Nxe5! Bxd1?? 7.Bxf7+ Ke7 8.Nd5# 1-0
</code></pre>

