If you claim to have the Assassin, you can perform the Assassinate action. Pay 3 coins to the supply and choose another player. That player must choose and lose one of their cards.
This action can be counter-acted if the target (but not another player) claims the Contessa. Note that if the claim is challenged and the target loses the challenge, the target will lose two cards (one for losing the challenge and one for the Assassinate) and so will definitely be out of the game.
If an Assassinate action is successfully counter-acted, you still have to pay the 3 coins. But if it is challenged and you lose the challenge, you don't have to pay the 3 coins.