Fool's Mate

Black checkmates White in TWO moves :

1.Pf3 e5    2.Pg4 Qh4#

It is not a good idea for White to move these pawns so soon, opening up the King to be attacked.

Scholar's Mate

White checkmates Black in FOUR moves..

The white Bishop and Queen attack the weak f7 square. This square is weak because the pawn on that square has only got the King to protect it.

1.Pe4 Pe5   2.Bc4 Bc5   3.Qh5 Nf6   4.Qxf7#

Scholar's Mate is probably the most common trap a beginner falls into.