The opposable thumb is the digit separating human beings and other primates from most of their other animal companions. It has been speculated that the hand-eye coordination made possible by the combination of stereoscopic vision and a grasping hand allowed primates to exploit their arboreal habitats in a fashion far more efficient than otherwise.
"Opposable" in this context refers to the ability to bend the thumb in opposition to the other fingers -- it can be brought face-on together with each of the four fingers on the hand. This inwards motion allows the hand to grasp to a far greater degree than fingers alone allow -- this is visible watching a baby or young child attempting to grab at something using only its fingers.
Many everyday actions would be nigh-on impossible without an opposable thumb: something as simple as picking up a sheet of paper from a desk requires either scrunching it up under the hand, or sliding it to the edge of the table. This is clearly impractical. Writing, or even using simple tools, would become a great difficulty.
The opposable thumb is clearly of great importance for human development; what of other primates? Most primates, and other animals, with opposable thumbs are primarily tree-dwelling or -climbing: a grasping thumb is supremely useful for holding onto branches and other such protrusions, and also for carrying something (fruit, leaves) back down with you.
Many different animals possess opposable thumbs, such as:
1 Yes, pandas are (usually) now classified as members of the Ursidae family (i.e. bears): see http://en.wikipedia.org/wiki/Giant_Panda.