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

[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:

[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:

[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