A "mulligan" is a decision made by a player to return his hand to his library for an opportunity to draw a new one after shuffling. 
The current DCI-approved mulligan is the "Paris mulligan", where the player may mulligan for any reason, without revealing his or her hand, and then draw one fewer card than was in the returned hand. A player may repeat this process as many times as he desires and at any time.
All supported multiplayer formats support one "free" mulligan, that is, the first mulligan for each player during each game does not require them to draw one fewer card. This rule was instituted for Two-Headed Giant and later adopted for all multiplayer formats.
One card from Darksteel, Serum Powder, interacts with mulligans.
From the Comprehensive Rules
- 103.4. A player who is dissatisfied with his or her initial hand may take a mulligan. First, the starting player declares whether or not he or she will take a mulligan. Then each other player in turn order does the same. Once each player has made a declaration, all players who decided to take mulligans do so at the same time. To take a mulligan, a player shuffles his or her hand back into his or her library, then draws a new hand of one fewer cards than he or she had before. If a player kept his or her hand of cards, those cards become the player's opening hand, and that player may not take any further mulligans. This process is then repeated until no player takes a mulligan. (Note that if a player's hand size reaches zero cards, that player must keep that hand.)
- 103.4a If an effect allows a player to perform an action "any time [that player] could mulligan," the player may perform that action at a time he or she would declare whether or not he or she will take a mulligan. This need not be in the first round of mulligans. Other players may have already made their mulligan declarations by the time the player has the option to perform this action. If the player performs the action, he or she then declares whether or not he or she will take a mulligan.
- 103.4b In a multiplayer game, the first time a player takes a mulligan, he or she draws a new hand of as many cards as he or she had before. Subsequent hands decrease by one card as normal.
- 103.4c In a multiplayer game using the shared team turns option, first each player on the starting team declares whether or not he or she will take a mulligan, then the players on each other team in turn order do the same. Teammates may consult while making their decisions. Then all mulligans are taken at the same time. A player may take a mulligan even after his or her teammate has decided to keep his or her opening hand.
- 103.4d The Commander casual variant uses an alternate mulligan rule. Each time a player takes a mulligan, rather than shuffling his or her entire hand of cards into his or her library, that player exiles any number of cards from his or her hand. Then the player draws a number of cards equal to one less than the number of cards he or she exiled this way. Once a player keeps an opening hand, that player shuffles all cards he or she exiled this way into his or her library.
The original mulligan rule applied only if a player had an initial hand of seven lands or no lands. He or she would then have the option to reveal his or her hand, shuffle it back into the deck and then draw a new hand of seven cards. Those cards would become his or her opening hand (the process could not be repeated). This was replaced by the current mulligan system called the "Paris" mulligan. The Paris mulligan was named after the second constructed Pro Tour to use this system in April of 1997 (the first being Pro Tour L.A. sometime before).