Mayet’s Mate is a checkmate pattern where a rook is placed right next to the enemy king whilst supported by a distant bishop. The king is further obstructed by his own pieces.
Mayet’s Mate Example 1

Diagram above: 1.Rd8# demonstrates the checkmate pattern. The rook is supported by a distant bishop whilst the enemy king is also obstructed by his own pawns.
Mayet’s Mate Example 2
Aron Nimzowitsch used this checkmate pattern in his game against Semion Alapin, played in Saint Petersburg, 1914:

Diagram above: Nimzowitsch played 1.Qd8+. He sacrificed the queen to open the e-file for the rook on e1. Black is forced to play 1… Bxd8. Mayet’s Mate is coming on the next move.

Diagram above: 2.Re8# is checkmate. The white rook is supported from a distance by white’s bishop on c6, whilst the rook itself checks the enemy king and covers e7.
Interesting Notes on Mayet’s Mate
Mayet’s Mate is very similar to Anderssen’s Mate (where the rook is supported by a pawn instead of a bishop). It can also be confused with the Opera Mate (where the bishop plays an even more important role).
Difference Between the Opera Mate and Mayet’s Mate

Diagram above: There is a subtle but important difference between Mayet’s Mate and Opera Mate. In the case of the latter, the bishop plays an even more important role by covering the square in front of the enemy king.