What are some of the formats on here mean?

[jaceperry]

I know what standard is and very little about modern and legacy but what are the others? I know nothing about paupers or others like it.

[FaustCrowley]

I'm sure there are some FAQs kicking around here and I would have gone directly to Wizards but the link was not working for me. Here is the Wiki:

https://en.wikipedia.org/wiki/Magic:_The_Gathering_formats

[Crusader]

Vintage, Legacy, Pauper and EDH (commander) are what you call eternal formats. Meaning you can pretty much use cards that date back to Beta. Vintage is the most powerful and fast paced format in the game. Vintage uses a restricted list (only one copy of the card can be used) for cards that are over powered. Legacy is the second most powerful and fast paced format but it has a banned list instead of a restricted list. You will notice more than half of cards on the vintage restricted list are banned in legacy. In vintage you can expect games to last 1-3 turns, while legacy can be anywhere from turn 4-5 kills at average, though some decks combo off turn one for the win in legacy. Pauper is another Eternal format in which you can use only common cards, and Paper pauper is apperently not sanctioned by wizards of the coast. The MTG Online pauper format has a slightly different environment than paper pauper as far as card availability, and some cards have different rarities. Local Play groups have been known to play different variations of pauper such as standard or modern pauper. In modern the card availability starts around Eighth Edition to the current as sort of a new beginning, new format, new child of MTG. Though don't be deceived some newer card prints are not allowed in Modern. Special Sets like Conspiracy_, commander, Planechase, Promos have reprints of cards that are not in modern. Being fluent with modern requires obtaining a vast knowledge of the cards and sets in the game.

[CommanderMeanderer]

Since Commander was only mentioned briefly, I can flesh out the format for you. Commander is a 100 card singleton format, meaning that you can only have one copy of a card in the deck (excluding basic lands). It is typically a 4 person free for all game, and is a social format with some interesting politics. Each player starts at 40 life instead of 20. Each deck has a commander, which counts as one of the 100 cards. The commander has to be a legendary creature, and stays in a unique zone called "the command zone". You start off the game with your commander in that zone, and can play them anytime you could play a creature just like it was from your hand. If you commander dies, is exiled, or shuffled back into your library, you may choose to put them back in the command zone. If you do, you have to play a commander tax for each time the card has returned to the command zone, making the creature 2 colorless more to cast each time it returns to that zone. If you're dealt 21 or more damage from a single commander, you automatically lose the game. Your commander determines what cards you can have in the deck too. The commander's color identity is determined by the mana symbols on the card, meaning both it's casting cost and activated abilities. Therefore, commanders like Yasova Dragonclaw would be able to play cards with green, red, and blue mana symbols on the card. Games can easily last 1-3 hours. 

You might see people talk about French Commander, Elder Dragon Highlander (EDH), pauper commander/PDH, or Tiny Leaders. French Commander is played 1 v 1 instead of 4 player free for all, and has its own ban list separate from regular commander. EDH is the same exact term for commander, named after the legendary elder dragons, but Wizards changed the name to officially be Commander back in 2011 (if memory serves me right). Pauper commander is not a sanctioned format, but is just a casual format. Usually this means you have an uncommon and non-legendary commander and only commons in the other 99 cards. Tiny Leaders is a new commander variant, following all of the rules of regular commander, except every card must be converted mana cost 3 or less. The decks are only 50 cards, and you start at 25 life instead of 40. There is no commander damage since you start at a lower life total.

TLDR: it's a 4 player free for all 100 card singleton format with the command zone/a commander.

Sorry for the huge description, but it's a very different format with a lot of variations. Hope I helped!