You can trade pokémon, and attempt other stuff that I haven't covered here.
Trainers can battle one another, either by arrangement or ambush. In such cases, their pokémon fight one another. Both trainers describe their attack (either openly or in PM) and the pokémon act simultaneously. If there are more than one pokémon involved in a battle, trainers have to declare which of their pokémon is attacking which other pokémon and these attacks are rolled separately. All the opponents other pokémon in a battle must be attacked (so no ganging up). There must be even numbers of pokémon on both sides of a battle (excess pokémon won't be counted— they like to play fair even if trainers don't).
Whoever gets the highest roll (after modifiers are taken into account) is victorious. In multi-pokémon battles, the separate rolls are added up.
In a pitched battle, and outcome of a win or loss is decided amicably by the trainers before hand. In an ambush, the victorious trainer has time to steal one pokémon before getting away. They needn't steal anything, and indeed may decide not to. Deep-grained taboos against leaving people defenceless mean you can never steal a player's last pokémon.
If attack decisions aren't sent in within 36 hours, they'll be automatically decided. Fleeing is possible in a trainer battle.
Stolen loyal pokémon maintain their old loyalties. Pokétrainers have to be found to be ambushed, which requires a successful finding roll.