Examples

An x-ray capture is a winning trade that uses an x-ray.
This is easiest to explain with some examples:

It looks like Black's queen on f5 is only attacked once by White's queen.
However, there is an x-ray of White's rook on f1.
Indeed, after White captures on f5 with 1.Qxf5, the queen of f5 is x-ray protected by the rook on f1.
After Black recaptures with 1...Rxf5, there is no longer a rook on f4, and White can win the rook and the game with 2.Rxf5.

X-ray captures can also happen "through" pawns:

Black just played Ra1, threatening to capture the rook with Rxd1 as well as promoting the pawn with b1=Q.
The rook on a1 is only attacked by the White's rook on d1. However, there is an x-ray by White's bishop on g7. Therefore, White can win with the x-ray capture 1.Rxa1. The rook on a1 is now x-ray defended by the Bishop.
After Black recaptures with 1...bxa1=Q, there is no longer a blocking pawn on b2, and White can capture the new queen with 2.Bxa1.
White is two pawns up in the resulting endgame and should win.