In the modern nation state, courts ("justice") operate with the threat of violence. The exact way a dispute is settled varies (jury, single judge, etc.), but the "finality" of the resolution they provide is based on the fact that the state asserts the final say, and if you don't agree, it doesn't matter, because they have men with guns. Importantly, they don't just have some men with guns (that's something you might have, too), they have an overwhelming number of men and increasingly large guns depending on how much you don't want to accept their say.

If you are thinking "that's a stupidly over-simplified way of thinking about modern justice systems", then that's fine; a lot of what follows is precisely about the finesse around this simplified view of the world.