We don't know how they work.
Anticonvulsants were originally tried on mood disorders based on a kindling phenomenon that has been observed with regard to seizures. First, if you stimulate the brain with regular pulses of electricity, you get repeated seizures after each shock. Then, after several such electrically stimulated seizures, you get seizures occuring on their own. In other words, after a while of this stimulus then seizure sequence, a point is reached where even if there is no electrical pulse supplied, the brain seizes spontaneously on its own. The earlier seizures are said to have kindled the later ones.
You grow up in a troubled home, you get depressed. Your parents get divorced, you get depressed. You lose your job, you get depressed. You get divorced yourself, you get depressed. Suddenly a point hits where you start getting depressed even though things are going well for you, out of the blue, for no reason. The earlier event then depression sequences have kindled the later out of the blue spontaneous depressive episodes.
This symmetry between electrical pulses and seizures on the one hand, and environmental stressors and mood disorder episodes on the other, is one reason that research psychiatrists began to think anti-seizure medications known as anticonvulsants might also be helpful for mood disorders as well. Tegretol, Valproic Acid, and Lamotrigine are examples of these agents, i.e. anticonvulsants that have been successfully tried to treat mood disorders.
On a chemical level, one theory is that anticonvulsants work, at least in part, by being pro GABA current which is a neurotransmitter current that tells the brain to relax, and/or by being anti Glutamate/Aspartate currents that tend to get the brain excited. By either stepping on the brake or cutting back on the gas, these agents may help stabilize things.
Lithium is a mood stabilizer too but it is not an anticonvulsant and works by other means. Lithium does lots of things and is an effective treatment but we are just early in teasing out what is the wheat and what is the chafe here.
Mood stabilizers are primarily used for bipolar disorder.
Atypical Antipsychotics are not classified as mood stabilizers per se, but they are useful in the treatment of bipolar disorder.
Thinking of baseball, with nerve cells you can think of a pitcher nerve cell throwing balls of neurotransmitters, i.e. Serotonin balls to a catcher nerve cell. When the catcher cell catches the pitch in his catchers mitt, changes occur within it. There is a lot of focus on these changes within the catcher cell in bipolar disorder. One of the more interesting ideas is that mood stabilization, regardless of the agent used, is acheived via these changes.
For Technically inclined readers:
Pitcher Cell= Presynaptic Neuron
Distance between pitcher cell and catcher cell= synapse, the microscopic spaces between nerve cells where neurotransmitters are secreted.
Catcher Cell=Postsynaptic Neuron
Catcher's Mitt=Postsynaptic Neuron Neurotransmitter Receptor
Changes within catcher cell= "second messenger systems" very complex chemical processes that occur in the postsynaptic cells after medications are introduced. Gene expression, nerve cell growth, and calcium transport may be involved in this process. A current area of intense research involves Protein Kinase C (PKC). Overactive PKC may be part of the problem in the manic part of manic depressive illness. Lithium and Valproic Acid seem to calm PKC activity and stabilize manic behaviors, and it is possible that people inherit a defective modulator of PKC activity that leaves them vulnerable to bipolar problems.