Common to all MMOs is the concept of "Hate." It's called "Enmity" in Final Fantasy XI, and learning to manage it is probably the single most important concept in the game. From the lamest little battle you'll fight, to the epic struggles against final bosses, managing Enmity is essential to your success.
When fighting monsters as a group, every action a player takes angers the monster more and more. If one player angers the monster significantly more than another, the monster will turn its attentions onto that player. If the player is a mage or other job with low defense, this is a very bad thing - they're going to be taking a dirt nap in seconds.
"Tanks" - or "Shields," as the Japanese players on Final Fantasy XI refer to them - are solely tasked with managing Enmity and keeping it focused on them. Using abilities specifically designed to draw a monster's attention, they protect other players from pissed off monsters and minimize the number of targets the healer has to focus on. On top of that, some jobs have abilities to manipulate or even get rid of Hate, making the tank's job easier.
The game keeps track of each and every player's Enmity, and if you're in a party, the game keeps track of its Enmity too. Generally speaking, every point of damage dealt to a monster or healing lavished on a friend, translates into one point of Enmity for the player. The idea, then, is for other players' total Enmity to never exceed that of the tank. However, since everyone's Enmity adds up, if one player deals too much damage too quickly, the tank may not be able to regain the monster's attention before it's too late.