Named for its restriction to only allow common cards, Pauper had its origins on Magic Online where it was first popularized by players, but quickly gained a big following and became an official Constructed format. Only cards released at common rarity are legal. Other than that, the usual deck construction rules apply: up to four of any one card can be played, except for an unlimited number of basic lands.
Pauper is often combined with other format restrictions, like Pauper Standard or Pauper Modern. But the classic Pauper format allows commons from all of Magic's past and the resulting environment features various high-powered decks, including the full spectrum from beatdown to control to combo strategies.
Pauper is no longer confined to Magic Online. When played with physical cards the format is often called “Paper Pauper.” Some differences in card rarities between the online and offline world exist, and some communities have opted for an expanded ban list because of it. Also, for some of Magic's older card sets it's hard to tell how rare a card actually is.
At the MKM Series Pauper side events, we follow the official rarity designations listed at: http://gatherer.wizards.com/
We also use the official list of banned cards listed at: http://magic.wizards.com/en/game-info/gameplay/formats/pauper
As long as a card has been printed at common rarity, any version of the card is legal in Pauper. For exanple, Counterspell was printed as an uncommon in Fourth Edition and as a common in Fifth Edition. Both printings are legal in Pauper.