This scenario is often referred to as a discovered attack (or check) with capture.
Less often, a move may discover multiple attacks, as in the first diagram where the knight's departure opens two crisscrossing diagonals.
The result is a double check: One check is given by the rook, discovered by the capturing pawn's move; the other by the bishop, created by the captured pawn's removal.
Such a check is extremely rare in practical play, but it is sometimes found in problems.
The diagram illustrates a trap in the Advance Variation of the French Defence, based on a discovered attack.