Which chess piece can only move diagonally?

The bishop, and the consequence of moving only diagonally is that each one is stuck on its starting colour for the whole game. You begin with one light-squared and one dark-squared bishop, and if you lose one, half the board becomes permanently unreachable by the other - which is why a bishop pair is worth noticeably more than twice a single bishop.

The piece was not always a bishop. In the Indian original it was an elephant, and it kept that name across the Islamic world and into Russian and Spanish; the English bishop and the French fou, the jester, are local reinterpretations of a shape nobody recognised. It also used to be far weaker, jumping exactly two squares diagonally, until the great rules change of about 1475 that also gave the queen her modern power and turned a slow game into a fast attacking one.

The colour-binding produces one of the tidier endgame facts: a king and a single bishop cannot force checkmate against a lone king, no matter how well played. Neither can a king and knight. It is a rare case in a game of nearly unlimited complexity where you can prove that something is simply impossible.