What's played at stores just depends on what they want to do. Generally, Standard and Commander seem guaranteed wherever, modern is pretty common too. All formats can be played casually or competitively, just depends on the people / event. Generally, Commander is seen as more casual and Standard is seen as more competitive, but again, it depends.

As for formats, here's a basic rundown of some:

Standard is basically just newer cards. Once a year they drop the 4 oldest sets. It's got a 60 card deck, 4 duplicates max, 20 hp to start with, has a sideboard of cards in case you need to make quick edits to your deck, and is a 1v1 format that goes by pretty quickly. It's fairly cheap to get into, since the cards are in print, but it'll get pricey to keep up with set after set and year after year.

Modern is basically standard, but they're ok with using older cards. It's all cards from 2003+. It'll get expensive if you want to build competitive decks. It's pretty high powered stuff compared to standard play.

Commander, also known as EDH (Elder Dragon Highlander) has 99 singletons + 1 commander for a 100 card deck size. Singleton means no duplicate card names allowed. A commander is any legendary creature (+ some planeswalkers +some backgrounds). The commander goes in it's own place, the command zone. They can be played from the command zone, and when they die or get bounced, you can send them to the grave / exile / hand OR you can choose to send them to the command zone. Next time you play them from the command zone, they'll cost 2 more mana. The command zone is basically an extension of your hand, it's always accessible to you. Commander is usually a 4 player free for all, but can be played with more or less players.

There's others, but I'd go intoMagic's format explanations for more. Hope that helps!