I know that the rule states "The game is drawn upon agreement between the two players. This immediately ends the game." but what I'm interested in knowing the exact way to offer a draw because I've never played at a club (always played online, where you simply click the 'Offer draw' button). So do you actually say "Draw?" or "Do you extend your hand after you play your move?" Just wanted to know the norm.
In tournament play, you make your move on the board, state your offer clearly (I say "I offer a draw" and make eye contact to make sure I am understood), then press your clock.
To add to the answer posted by @dfan (who is correct on all counts), the draw offer is only good for that move. If you make a move, then offer a draw, and your opponent makes a move, then the draw offer is implicitly declined.