The most popular format: social, usually with 3-4 players.
Deck
100 cards, every card unique (singleton), 1 legendary creature as commander.
Card pool
Almost every card ever printed.
Notable
You start with 40 life; cards may only be in the colors of your commander (color identity).
Rules that differ from the basics
A 100-card singleton deck: aside from basic lands, every card may appear only once.
You choose exactly 1 legendary creature as your commander; it starts in the command zone.
Color identity: your deck may only contain cards in your commander's colors (mana symbols in the card text count too).
You start with 40 life instead of 20.
With more than 2 players, your first mulligan is free: you don't put a card on the bottom for it.
Commander damage: if you take 21 total combat damage from a single commander, you lose immediately.
Command zone: you may cast your commander from there. Each time after that it costs {2} more (commander tax). If it dies or gets exiled, you may send it back to the command zone.
Usually played with 3-4 players, so lots of negotiating and alliances.
The easiest way to play casually with friends. Ready-made "precon" decks make getting started simple, and from there you slowly upgrade. Games last longer and revolve around fun, memorable moments instead of pure winning.
Almost anything goes: extremely powerful and expensive.
Card pool
Virtually every card ever printed, with a banlist.
Notable
Iconic expensive cards (including the original dual lands).
For enthusiasts who want the most powerful 1-on-1 games and are willing to invest. Often played with "proxies" (fake cards) in casual circles to keep costs down.
Like Commander, but with a planeswalker and smaller.
Deck
60-card singleton, a planeswalker as your "oathbreaker" + 1 fixed "signature spell".
Card pool
Almost every card ever printed.
Notable
You start with 20 life; faster and cheaper than Commander.
Rules that differ from the basics
A 60-card singleton deck.
You choose a planeswalker as your "oathbreaker" plus one instant or sorcery as your "signature spell"; both start in the command zone.
You may only cast your signature spell while your oathbreaker is on the battlefield.
You start with 20 life.
The command zone works like in Commander: {2} extra each time you recast it.
A fun alternative to Commander if you love planeswalkers and want shorter, cheaper games.
MTG Arena also has digital-only formats: Alchemy (with digitally rebalanced cards), Historic and Timeless (large Arena card pools), and Brawl (Standard-singleton with a commander). You don't play these with paper cards.