SSRIs are a class of anti-depressants. The initials SSRI stand for "selective serotonin reuptake inhibitor." They act within the brain to increase the amount of the neurotransmitter, serotonin (5-hydroxy-tryptamine or 5-HT). Many drugs in this class are familiar through advertising. This class includes Prozac®, Zoloft®, and Paxil®.
In the brain, information if passed between two neurons (nerve cells) via a synapse, a small gap between the cells. The neuron that sends the information releases neurotransmitters (with serotonin among them) into that gap. The neurotransmitters are then recognized by receptors on the surface of the recipient cell, which upon this stimulation in turn relais the signal. About 10% of the neurotransmitters are lost in this process, the other 90% are released from the receptors and taken up again by the sending cell (thus reuptake).
Depression can be caused by a lack of stimulation of the recipient neuron at a synapse. To stimulate the recipient cell, SSRIs inhibit the reuptake of serotonin. As a result, the serotonin stays in the synaptic gap longer than it normally does, and has the chance to be recognized again (and again) by the receptors of the recipiant cell, which can finally be stimulated that way.
Why not give serotonin directly? First, pure serotonin would turn on every synapse it reaches, whereas SSRIs only enhance a signal that is already present, but too weak to come through. Second, the first letter of SSRI standing for selective indicates that not all synapses are affected, but only selectively those that are responsible for the the mood and, thus, the depression.