Besides single battles, there are also Double, Triple and Rotation battles. In these battles, two or three Pokémon are on the field at once, for each side. They each have a variety of special rules.
In these battles, it's obviously 2v2. The first two Pokémon in each player's party are sent out. The actions of the two Pokémon on both sides are chosen before anything happens.
This can include using items or switching with other Pokémon. Since you have two actions per turn, you can use items, switch Pokémon out twice (and more); in a single turn. Using moves now also requires you to select your target. Moves can have a number of target types and you can target both team mates and opponents.
After all Pokémon have performed an action, any Pokémon that fainted during that turn will be replaced by another Pokémon - not immediately after they faint, but before anyone else performs their action. Other than this, battles play out in a similar way to Single Battles.
In these battles, three Pokemon from each side are used; totalling six Pokémon on the battlefield. Battles work just like Double Battles (except each side has an extra Pokémon to use). The two main things consider here (and one's an extension of the other) are Range and Pokémon placement.
Most moves have a Range extending only to those adjacent to them (including diagonally). In Triple Battles, Pokémon on the left or right side of the group can only target the enemy on their side, the middle enemy and the middle ally.
Meanwhile; the Pokémon in the middle can target all other Pokémon, both enemies and allies. If a move is 'Long Range', they can target all other Pokemon, no matter where they are positioned on the field.
Placement is important here. You should position Pokémon in a way so that they're up against enemy Pokémon that their type is strong against. Obviously - in the case of the middle Pokémon - they should be hardier than the other Pokémon and should be at least type-neutral to all three enemy Pokémon. If their type's strong against them, all the better.
There is also one extra type of action that can be performed in Triple Battles, called 'Moving'. A Pokémon, instead of using a move, item etc; can opt to switch with an adjacent teammate. When it reaches their turn; the two Pokémon switch places.
If the other Pokémon that moved then performs a move afterwards and the target is now out of range, the move misses entirely, so be careful with it. If there's no Pokémon to switch with (ie. there are only two or one that haven't yet fainted in the entire party), they will simply move into the empty position.
The black sheep of battle-types, Rotation Battles work similarly to Single Battles, but have three Pokémon on the field at once (per side). In these battles, there is one 'active' Pokémon on each team; with the others standing to the side. During each turn; only this Pokémon performs an action. What differs this battle-type from others is the ability to rotate.
Rotating occurs before any moves are used. Neither side knows the Pokémon that the other side will rotate to, if they rotate at all, meaning that it's all about prediction.
Obviously, using a range of Pokémon types will be of advantage here; so that you can switch to a strong type whenever the enemy rotates. If you have one type that's strong against the types of all of the enemy Pokémon, this should be used all the time.
To rotate, simply select the ' Fight ' option, then press the rotating arrows on the bottom left and right corners of the bottom screen to switch to the other Pokémon. When you do, select a move to use for them.
If an item is used the currently active Pokémon will stay active. If a Pokémon is switched with one not on the field; the currently active Pokémon is the one that's switched out. Keep these things in mind when using items or switching Pokémon.