I'm going to answer this question in terms of how I understand what you're asking.
First of all you two players can play:
- Local battles
In local battles players can set the rules however they see fit, they can allow/disallow Special Pokemon and can set the number of Pokemon that can be used.
- Battle Spot
Two players can also battle online via Battle Spot (accessed from the Festival Plaza in SM.) There, players have a choice of doing single battles or double battle. Players can decide weather they want to play ranked or just friendly battles.
In regards to the styles/methods/tiers two players can battle against each other:
You have official formats, such as VGC.
There are Single Battles, Double Battles. (In the past Triple Battles, Rotation Battles and Launcher Battles existed as well.)
Also, on another note: I don't know if Inverse Battle Multiplayer can be done.
There are also Smogon tiers, two players can battle each other with. The tiers aren't officially supported by the game, but players can play using the tiers for fun or a new challenge. These tiers include the likes of:
There are also seasonal/monthly competitions with unique rules (at times) which can be entered through Battle Spot's Live Competition function. (Live Competition battles are another type of battles- this may include Single Battles and Double Battles.) An example of a competition with unique rules is the Pikachu Cup.