Mill is a term used to describe the act of putting cards from a deck into another zone to deplete the card in the opponents library.
Milling can be a successful strategy due to the fact that a player who can not draw a card from his library when he is required to (due to the library not containing any cards any more) loses the game.
Cards that mill the opponents deck normally put those cards into the graveyard, but exiling them or putting them into any other zone in the game can be counted as milling aswell. This makes milling cards useful for planing ahead, since the milled cards are often made unusable, giving the defending player fewer options. Also, the graveyard is a public zone, allow the mill player to see what he got rid of and making it easier to calculate what options the defending player still has available.
The term itself was coined from an artifact card called Millstone, which was released way back in Antiquities. The entire idea of depleting the opponents library was a very artifact based strategy in the early sets of the game. Milling is primarily a blue play style, aided sometimes by black, especially in the House Dimir, a Ravnican guild. Milling has increased in popularity over the years, and the card Millstone itself has been reprinted many, many times over, all the way into 2014 Core edition.
Even up to the Mirrodin block, Milling was ruled by artifacts. Milling didn't really become primarily blue until Tenth Edition. This really set the standard for mill players, and finally gave blue a serious defining power, rather than just general control.
However, due to the difficulty of milling enough cards and the circumstance that milling neither provides direct card advantage nor influences the board situations, milling is rarely seen as a tournament strategy outside the realm of Combo, e.g. decks that win by Brain Freeze.