Hyper Offense: Usually 1 hazard setter and 5 powerful sweepers, Hyper Offense's purpose it usually to set up hazards and sweep as quickly as possible.
Balance: Just what it sounds like, Balance teams are generally balanced between Walls and Sweepers, and other roles.
Example of a Hyper Offense team
Stall: Mainly 5 Walls and 1 powerful sweeper, Stall uses Status effects and weather to wear down an opponent's HP, as well as phazing with hazards up, and essentially dragging the battle out, frustrating the opposing player and goading him to make a hasty mistake, then bringing in the sweeper (usually a scarfer) to finish the game.
Example of a Balance team
Volt-Turn: Generally offensive, Volt-Turn teams use the moves U-turn and Volt Switch to build offensive momentum and sweep, and almost always contain a spinner.
Example of a Volt-Turn team
Trick Room: Mainly slow and powerful Pokemon such as Conkeldurr or Scizor, operating under Trick Room, set by bulky tanks. Very under-rated style, as most teams are meant to be as fast as possible, and when under Trick Room, tend to fall apart rapidly.
Example of a Trick Room team
4Drag2Mag: 4 Dragons because they are extremely powerful sweepers, and a Magnezone and Magneton to trap Steel types and tank Ice type moves. Also a very under-rated playstyle.
Example of a 4Drag2Mag team
Baton-Pass: Baton Pass teams are rather difficult to pull off, but center around the move Baton Pass. Most Baton Pass teams are also fairly the same as there is not an abundance of Pokemon that learn Baton Pass. However, the idea is to build up some stat boosts with 1 Pokemon, then Baton Pass it to your next Pokemon, which will set up a Substitute with the aid of the stat boosts it has gained, then Baton Pass that out to a sweeper such as Scizor, which will then set up a Swords Dance and sweep. Smeargle, Ninjask, Vaporeon, Espeon, and Volcarona are all very common in Baton Pass teams. Baton Pass teams are easily ruined by moves such as Roar, Whirlwind, and Circle Throw, and if any of the Pokemon are killed the stats are lost, and when you lose the accumulated stat boosts before they get to their destination, you usually lose. But don't get me wrong, a good Baton Pass team is utterly lethal when played correctly.
Example of a Baton Pass team
Also note that the more basic styles of teams can be run in weather, which is arguably a playstyle of it's own.