The Scholar's Mate is a simple attack similar to the Fool's Mate. Both are quick openings which rely on the queen attacking along a diagonal, resulting in checkmate. In the Fool's Mate, the queen wins the game by itself, but in the Scholar's Mate, it's supported by a bishop.
The description below is accurate for a white mate, although black has several moves that could vary, such as moving any of the queen-side pawns instead at any point after the first move.
1. e4 e5
2. Bc4 Bc5
3. Qh5 Nf6??
4. Qxf7++
For those who don't read chess notation, we'll play it from the white side and I'll try to spell it out. First, get the king pawn out of the way to E4 (two spaces forward). Then, move the white bishop to C4 to cover the black pawn diagonally to the right of the black King (F7). Next, get your queen out to H5, on the far right of the board, on the diagonal with the black king. Your next move is to capture the pawn covered by your bishop with the white queen for mate.
Right before the mate, white is in caps:
8 | r | | b | q | k | | n | r |
+---+---+---+---+---+---+---+---+
7 | p | p | p | p | | p | p | p |
+---+---+---+---+---+---+---+---+
6 | | | n | | | | | |
+---+---+---+---+---+---+---+---+
5 | | | b | | p | | | Q |
+---+---+---+---+---+---+---+---+
4 | | | B | | P | | | |
+---+---+---+---+---+---+---+---+
3 | | | | | | | | |
+---+---+---+---+---+---+---+---+
2 | P | P | P | P | | P | P | P |
+---+---+---+---+---+---+---+---+
1 | R | | B | | K | | N | R |
+---+---+---+---+---+---+---+---+
a b c d e f g h
Now, capture the pawn next to the black King, and voila:
8 | r | | b | q | k | | n | r |
+---+---+---+---+---+---+---+---+
7 | p | p | p | p | | Q | p | p |
+---+---+---+---+---+---+---+---+
6 | | | n | | | | | |
+---+---+---+---+---+---+---+---+
5 | | | b | | p | | | |
+---+---+---+---+---+---+---+---+
4 | | | B | | P | | | |
+---+---+---+---+---+---+---+---+
3 | | | | | | | | |
+---+---+---+---+---+---+---+---+
2 | P | P | P | P | | P | P | P |
+---+---+---+---+---+---+---+---+
1 | R | | B | | K | | N | R |
+---+---+---+---+---+---+---+---+
a b c d e f g h
The white bishop prevents the king from taking the queen, and there's no black piece than can attack, either. Checkmate.
This move is easily defeated by black at pretty much any point by moving the black knight to h6, or by moving the black queen to f6, or by getting a pawn up to d5 before the white queen is in place. And those are just the easy options. It usually won't work against an experienced player, unless they're just not paying attention, but it can surprise someone every once in a while.
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