"...Team Battle makes the player pick five out of six party members (all six for Double Battle), and faces them off in one-on-one elimination rounds (two-on-two for Double Battle). This requires one of the highest strategy levels, as the player does not know what Pokémon they face in a round."

"The player wins a round if they knock out their opponent's Pokémon first. They must win three out of five rounds to win the battle (two out of three for Double Battle). The round is considered a draw if all the Pokémon on the field are taken out due to moves like Explosion or Self Destruct. If there is a draw between the number of wins, the Trainer who won the last match wins the battle."

This is how Bulbapedia describes Waterfall Colosseum's Team Battle format in the Gen 4 Wii game Pokémon Battle Revolution, which (with the possible exceptions of Triple battles and/or Multi battles) is my favorite battle format in any Pokémon game.
(I'm particularly fond of the Team Battle Doubles format.)

Like Triple battles (and unlike most battle formats in the core series handheld games), you use all 6 Pokémon in Double Team Battles (lol, "Double Team", and unlike most of Smogon's Singles and Doubles formats, there is no switching out your Pokémon.

Those "elimination rounds" mentioned above are freaking HARDCORE, and if your 2 (or 1 for Singles) Pokémon wind up against something that they can't handle, you don't have the luxury of being able to switch them out and do all of that back and forth rock paper scissors crap that can make a 6v6 Single battle last 100+ turns, which means that each and every one of your Pokémon need to be prepared to handle almost anything, although in Doubles having Pokémon that work well in pairs is also very important.
(Both of your 2 Pokémon will rely heavily on one another to cover each others weak points, to a much greater extent than in a regular Double battle.)

Like almost all of the battle formats in Battle Revolution, no Pokémon, moves, or held items are banned (which is cool IMO), although because Battle Revolution was originally only designed to be compatible with Diamond/Pearl Versions, certain forms of some Pokémon (such as Giratina's Origin Forme) aren't usable in the game, which kinda sucks.
(You can still use moves that were added to Pokémon's movepools in Platinum or HG/SS Versions, though.)

There is no "Species Clause" or "Item Clause", which means that you could even use 6 of the exact same Pokémon with the exact same held item (such as a team of Choice Scarved Kyogres or something), which is a bit stupid IMO, but overall, I think that the Team Battle format is one of the best battle formats that TPC (or whoever comes up with this stuff) has ever come up with, and I think that (with some tweaking) it would make an excellent PvP battle format, including in later gens.
(Imagine a Gen 7 Team Battle Doubles format, complete with Mega Evolution and Z-Moves!)

Although I have no idea if anyone else feels the same way as me about this, I would like to see some versions of the Team Battle Singles and Team Battle Doubles formats added to Pokémon Showdown (or some other Pokémon battle simulator), with at least a few of the following modifications, to make it more appealing as a PvP battle format:



If anyone here has any ideas about how I might make any of this a reality (such as how I might pitch these ideas to Pokémon Showdown or some other Pokémon battle simulator), would like to ask any questions about these battle formats, or would just like to share your own thoughts about this or other cool, somewhat obscure battle formats (the format of Emerald Version's Battle Arena comes to mind), please leave a comment.

Sayonara!