How the count works
The count is of moves by both players, so fifty moves means one hundred half-moves. Any capture or any pawn move resets it to zero.
Over the board the claim works like a repetition claim: write the move that completes the fifty on your scoresheet, or claim after it, then stop the clock and call the arbiter.
Under FIDE Article 9.6 the game is drawn automatically after seventy-five moves by each side without a capture or pawn move, unless the last move delivered checkmate.
Why it matters
The rule stops a player from playing on forever in a position they cannot win. It also sets a deadline in some theoretically won endgames: king, bishop and knight against king can take more than thirty moves to mate even with perfect play.
Computer tablebases show a handful of positions that need more than fifty moves to win with best play. FIDE briefly experimented with exceptions for these in the past, but today the plain fifty-move rule applies to everything.