Formats are pretty simple to understand, in all honesty.
-First you have Standard, which is a constantly rotating format where only the newest cards in print are played.
-Then you have Modern, where you can play anything printed from Eighth Edition and forward.
-After that is Legacy, which allows cards from all over Magic's history, save for some limitations
-And then Vintage, which allows everything legally playable, pretty much. And the limitations of Legacy are instead restricted to only being allowed to have one.
Those are the BIG competitive constructed (meaning you build your deck before you go to the event) formats.
Then there is Limited, where you get packs of cards and build a deck right then and there to play against others and their just-built decks.
And then there are the more Casual constructed formats. My personal favorie is commander. 100-card singleton deck (only one of each card except for basic lands), with one card helming it as the "Commander". The color identity of the commander determines the color of the rest of the deck. For example, if you use
Brago, King Eternal, then your deck can only have blue and white spells in it. There is WAY more to it, but that's a basic description.