Captured pieces can be dropped back on the board instead of moving a piece.
Rules
All the rules and conventions of standard chess apply, with the addition of drops, as explained below.
A captured piece reverses color and goes to the capturing player's pocket. At any time, instead of making a move with a piece on the board, a player can drop a piece from their pocket onto an empty square on the board.
For example, a check that would result in checkmate in standard chess can be answered in Crazyhouse, if the defender can play a legal drop that blocks the check.
Promoted but captured pawns are dropped as pawns.
Example

White to play, with two pieces in their pocket: a knight and a bishop. White can check the black king by dropping a bishop on c3.

Now black can block the check by dropping a knight on e5.

Final position, white to move.
Notation
An extension to the standard chess notation is used to record drops. Drops are notated by the piece type, where the piece type is blank for pawns, followed by an "@" symbol, then the destination square. For example, @d5 means "pawn is dropped on d5 from reserve".
Strategy
Pawns and knights increase in relative importance in Crazyhouse, while rooks, queens, and bishops decrease in relative importance. If a king is put in check by any of the latter three pieces, from two or more squares away, dropping a pawn next to the king becomes defensively useful. A knight, on the other hand, cannot be blocked by anything and its offensive value is more manifest. That piece can be used effectively to maintain a strategic influence over a region.