Starting position
The moves 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.c3 Nf6 5.d4 exd4 6.cxd4 Bb4+ 7.Nc3 Nxe4 8.O-O Nxc3 9.bxc3 — the Greco Gambit in the Giuoco Piano. White has given up the pawn on e4 and castled, and the pawn on c3 has taken the knight and attacks the bishop on b4. This is the Aitken Variation.
The opponent's mistake
9...Bxc3? looks profitable: Black is already two pawns up, and the bishop attacks the rook on a1. But Black's king is still in the center, on the open e-file.
How the trap works
10.Ba3! — the bishop hits f8, and Black can no longer castle short. White gives up the rook: 10...Bxa1? 11.Re1+ Ne7 — the knight blocks, but it is pinned to the king. 12.Bxe7 Qxe7 13.Rxe7+ Kxe7 — White has taken the knight and then the queen, which stepped into the pin itself. Next 14.Qe1+ — check, and the queen attacks the bishop on a1. After 14...Kd8 15.Ng5 Rf8 16.Nxf7+ Rxf7 17.Bxf7 c6 18.d5 cxd5 19.Qxa1 the queen takes the bishop in the corner. If Black answers 15.Ng5 with 15...Re8, it is mate: 16.Nxf7#.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.c3 Nf6 5.d4 exd4 6.cxd4 Bb4+ 7.Nc3 Nxe4 8.O-O Nxc3 9.bxc3 Bxc3 10.Ba3 Bxa1 11.Re1+ Ne7 12.Bxe7 Qxe7 13.Rxe7+ Kxe7 14.Qe1+ Kd8 15.Ng5 Rf8 16.Nxf7+ Rxf7 17.Bxf7 c6 18.d5 cxd5 19.Qxa1
Result
White has a queen against a rook and two pawns — a decisive advantage. If Black doesn't take the rook (10...d6), White moves it away with 11.Rc1, and the bishop on c3 is left undefended.
How to defend (as Black)
Don't take the pawn on c3: 9...d5 is stronger — the pawn attacks the bishop on c4. And after 10.Ba3 there is no need to take the rook: 10...d5 is stronger.