MAS - i know a little of how SSRIs work, but MB might be able to explain things better.
When we are depressed or anxious the levels of a hormone called seratonin drops in the brain, this then has the effect of causing low feelings and anxiety so it is a vicious circle. This is only one of the systems that go wrong in the brain in depression, but it is thought to have a major influence. Seratonin is a neurotransmitter, a chemical which is passed from one nerve cell to another. When the body requires seratonin, the pre-synaptic cell (the cell that does the talking at the synapse - synapse is a connection between to neurons) releases seratonin into the "space" between the two cells. The post synaptic cell then picks up the seratonin "message" and this causes further effects, on mood and other physiological processes. In order to pick up the seratonin, the post synaptic cell must have receptors on its surface that the seratonin binds to. The receiving cell will "make" enough receptors to cope with the level of seratonin released. Anything else is mopped up by the Seratonin Reuptake Receptors, sort of recycled. So, it becomes a vicious circle again, the less you produce the less receptors are made and the more is mopped up, etc etc, you end up making less and less.
SSRIs (selective seratonin reuptake inhibitors) work by preventing or slowing down the reuptake of seratonin at the synapse, thuse making it available for longer to the receiving cell, which in turn makes more receptors to cope, and hte feedback mechanism results in the body pumping out more seratonin. That is one of the reasons why they are thought to have long term effects, as once the balance is reset in the body, it sort of maintains itself.
Of course, depression isnt that simple and there are lots of other cells and chemicals involved, but seratonin (often called the happy hormone) is a big factor when things go wrong.
Importantly, seratonin production is involved in many physiological processes, not just mood, which is one of the reasons why there are a broad range of side effects ( i dont know what the other processes are off the top of my head).
I know im a bit crap at explaining things, but that is how i understand it, MB may have to put me right