Commander
Commander is a casual, multiplayer, 100-card format where each deck is led by a legendary creature (or sometimes planeswalker). It is by far the most popular casual format of Magic: the Gathering.
Brawl
Historic Brawl is a casual, 100-card format where each deck is led by a legendary creature (or sometimes planeswalker). All sets on MTG Arena are legal, but some individual cards are banned.
Limited
Limited is a format where players build decks from a very limited pool of cards. It is one of the most popular formats, and is usually played either as sealed or as an 8-player draft, followed by several matches of games.
Sealed
Draft
Cube
Cube is a casual, multiplayer format where one player builds a box of cards, usually between 360 and 1000 cards. Then all players draft cards from the pool and build decks from their drafted cards to play against each other. At the end of the draft, they return all cards to the owner.
Standard
Standard is a rotating format where the most recent sets are legal. It is the most popular competitive format in Magic: The Gathering, and most commonly played on MTG Arena.
Pioneer
Pioneer is a non-rotating, competitive, 60-card format where all sets from Return to Ravnica forward are legal.
Modern
Modern is a non-rotating, competitive, 60-card format where all sets from 8th Edition forward are legal. It is the second most popular competitive format in Magic: The Gathering.
Legacy
Legacy is a non-rotating, competitive, 60-card format where all sets are legal, but some ultra-powerful cards are still banned.
Vintage
Vintage is a non-rotating, competitive, 60-card format where all sets are legal, and the most powerful cards in Magic: The Gathering are legal but may be restricted.
Alchemy
Alchemy is a non-rotating, competitive, 60-card format where all sets are legal, but only cards that are not very good are legal. It is the most popular format for people who like to play with cards that are not very good, and it is also the most popular format for people who like to play with cards that are not very good.
Historic
Historic is a non-rotating, competitive, 60-card format where all sets on MTG Arena are legal, but some cards are banned. This is basically MTG Arena's analogue of Legacy.
Timeless
Timeless is a non-rotating, competitive, 60-card format where all sets on MTG Arena are legal, but some extremely powerful cards are restricted. This is basically MTG Arena's analogue of Vintage.
Explorer
Pauper
Pauper is a non-rotating, competitive, 60-card format where all sets are legal, but only cards printed at common rarity are legal. It is the most popular budget format, although some of the cards can be shockingly expensive.
Penny
Penny is a non-rotating, competitive, 60-card format where all sets are legal, but only cards that cost $0.01 or less are legal. It is the most budget format, and it is also the most popular format for people who want to play with increased card variety.
Oathbreaker
Oathbreaker is a casual, 60-card format where each deck is led by a planeswalker and includes a signature spell.