Magic: The Gathering formats can be broadly categorized into two types: Constructed and Limited.

Constructed formats:

In Constructed formats, players build their decks ahead of time using any cards from a legal card pool. Here are the main Constructed formats:

Standard: This is the most popular format for new players, as it uses cards only from the most recent sets. This means the format is constantly evolving as new sets are released. Standard decks are typically 60 cards with a 15-card sideboard.
  
Modern: Modern allows players to use cards from Eighth Edition forward, creating a much larger and more powerful card pool than Standard. Modern is a great format for players who enjoy strategic deckbuilding and fast-paced gameplay.

Pioneer: Pioneer is a relatively new format that uses cards from Return to Ravnica forward. It offers a middle ground between the constantly changing Standard and the larger card pool of Modern.
 
Legacy: Legacy allows players to use almost all cards ever printed in Magic's history, with a few exceptions. This format is known for its deep strategic complexity and powerful decks.
   
Vintage: Vintage is the most powerful format of Magic, using all cards ever printed with very few exceptions. This format is known for its fast-paced games and incredibly powerful decks.

Limited formats:

In Limited formats, players build their decks on the fly during the tournament using a pool of cards that they are provided. Here are the main Limited formats:

Sealed Deck: In Sealed Deck, players are each given six booster packs to build their decks from.

Commander:

Draft: In Draft, players take turns picking cards one at a time from booster packs passed around the table.
 
Commander: Commander is a multiplayer format where players build 100-card decks around a legendary creature commander. The format is known for its long, multiplayer games and its emphasis on fun and social interaction.
  
These are just some of the most popular Magic: the Gathering formats. There are many other variations and casual formats that can be enjoyed by players of all levels.