If you shout, you say something very loudly, usually because you want people a long distance away to hear you or because you are angry.

If you say that someone is in with a shout of achieving or winning something, you mean that they have a chance of achieving or winning it.

If you are in a pub and someone you are with says 'It's your shout' or 'It's my shout', they mean that it is your turn or their turn to buy a round of drinks.