A German word meaning "compulsion to move". When a player is in zugzwang, any move will lead to a disadvantage or worse. Here is a simple example of zugzwang with Black to play:
Black's only legal move is advancing b3 but that allows axb3#. Black would very much prefer to "pass" but unfortunately that's not how chess works!
The concept is mostly relevant in the endgame. Here is a practical example, a possible line in the game Steinle-Webb, Senden 2014, which even involves mutual zugzwang! Whoever's turn it is, loses:
With White to play, Black would win! Notice 1.Nd3?? allows Bg6# and so White has no moves after 1.d6 cxd6.
Sometimes zugzwang can have surprising results, as this position shows: