What is a battle format?
A battle format, or just "format", is the set of rules you use when doing competitive battles. It decides which Pokemon are allowed, whether the battles are singles or doubles, what items are banned, and many other things.
The format picker as seen on Pokemon Showdown and Pokemon Champions, respectively.
Which battle format to use is decided before the battle (e.g. when you queue for a match). Throughout history, there have been hundreds of different formats, and naturally they differ substantially. So it's important to make clear what format you're talking about!
Some formats are official - as in, they're included on the real games, like Champions - while others are controlled by the long-standing competitive battling community, Smogon. Modern competitive Pokemon is played on Pokemon Showdown (a battling simulator) or Pokemon Champions (the official competitive game).
What battle formats are there?
Modern competitive Pokemon has two distinct "ecosystems", revolving around Champions (which focuses mostly on double battles) and Smogon (mostly single battles). Specifically:
Champions Doubles aka VGC is the most popular ecosystem for double battles. Rules wise, it follows the Regulations set out by Pokemon Champions (Reg M-A, Reg M-B, etc), so you need to specify which Regulation you're talking about as well. The official Regulation changes every few months. Official VGC tournaments with prize money are played via Champions Doubles.
Smogon OU is the most popular format for single battles. It's part of the ecosystem of Smogon tiers (like Ubers, OU, UU, RU etc), but OU is the most important of them all. Its ruleset includes everything in the current generation of Pokemon (e.g. Scarlet/Violet aka Gen 9), adjusted by Smogon-wide clauses and anything that's banned to Ubers. OU is the basis for competitions like Smogon Premier League (SPL), World Cup of Pokemon (WCoP) and Official Smogon Tour (OST).
Different Pokemon and strategies are most effective in each format, because of the varying rules of each format. The state of strategy in a format can be called the meta (short for metagame). Building teams and strategies around the meta is one of the pivotal aspects of competitive Pokemon.
To choose a format to play, pick a Pokemon you like and research where it is viable. For example, if I like Lokix, then I might play Gen 9 UU because Lokix is very strong in that format. If you just want to play the most popular formats, stick to VGC and OU.
Pictures
(These pictures are from Gen 8 but should still illustrate the number of possibilities)
Battle format options as seen in Pokemon Showdown's teambuilder. Notice formats from all generations are available. "Official" formats and Smogon formats are available. Not all have permanent ladders.
Battle format options as seen in Sword/Shield via Battle Stadium > Download Rules. Highlighted is Battle Stadium Singles (BSS) Series 10.
