A mulligan, taken from the golf term, is a means of balancing out bad draws for your opening hand. The most common type of Mulligan these days is the 'Paris Mulligan', and it is the only legally supported method for sanctioned tournaments. When you draw your starting hand, you are free to shuffle it back into your deck if you are dissatisfied with it. Each time you reject your hand, you shuffle those cards back into your library and draw one less than you had before. This means that you can actually mulligan down to 0 cards, although this is inadvisable.
In the Two-Headed Giant format, each player may take a 'free' mulligan, meaning that they may draw a new hand of 7 cards. Subsequent mulligans after this, however, require one less card to be drawn each time, as normal.
Strategically, people tend to mulligan if your hand contains 0, 1, 6, or 7 lands. Too few lands means that you won't be able to play your spells, and too many means you won't have much to play in the first few turns. The cost for your spells is also an important factor to consider when performing a mulligan, as a card that costs 2 colored mana can probably not be played by turn 2 unless you have those two lands in your hand, and too many high-cost cards means you won't have much of an opening presence. If your deck's mana curve focuses more on 2 than 3 or higher, it may be worth the risk to accept a 1-land hand, but even that can be a risky proposition should you not get the second land when you need it. You'll also need to be more aggressive with your mulligans when you are taking the first turn, because you won't have as much time to get the lands you need.