Neurogenesis is the growth of new brain cells, and that it occurs at all is an important departure from the prevailing dogma of as late as a decade ago. Certain types of systemic and specific damage done to the brain by drugs, trauma, or stress, once thought permanent, are now thought to be reversable through growing new brain cells. Relatedly, the rate of neurogenesis varies: learning new things, for instance, increases it; being born into a rough life or to a stressed mother can decrease it; a state of chronic stress can more-or-less shut it down.
This discovery has paved the way for experiments involving neurogenesis: some findings have shown that growing new brain cells allows the brain to better learn and remember new things, but researchers have also linked it to mood, by giving antidepressants to rats who are depressed (they no longer act depressed, and later autopsies reveal they grew more neurons than other, depressed rats), and by using radiation to selectively kill new neurons and effectively stop neurogenesis in rats who were being given antidepressants (they went right back to being depressed). This data has allowed a rather audacious new theory, that 1). Depression is fundamentally a malfunction of neurogenesis rather than of brain chemistry, as was previously thought, and 2). Neurogenesis is nearly synonymous with brain plasticity, or the adaptability of one's brain.
To put this very simply, this hypothesis asserts that being happy (or at least, not depressed), continually growing new brain cells, and having an adaptable brain are all deeply connected enough to be considered the same thing. When there's a malfunction in any of these, it soon becomes a malfunction in all of them: for instance, when something occurs that hinders the creation of new brain cells, the brain loses some of its plasticity, which may cause stress or depression, which in turn hinders the creation of new brain cells, and so forth in a vicious cycle. Similarly, any therapy that helps one of these things- happiness, plasticity, and neurogenesis- helps them all (albeit with a one-month lag, the time it takes for new neurons to mature and connect to other neurons).
I heartily recommend Seed Magazine's special on neurogenesis and mood. Eero Castren also suggests an aggressive generalization of this idea- that "mood represents a functional state of neural networks" (accessible commentary by Derek Lowe). And, for the scholarly folks, here's a well-linked overview of recent activity in this corner of science from Wikipedia:
Malberg et al. (2000)  and Manev et al. (2001)  have linked neurogenesis to the beneficial actions of certain antidepressants, suggesting a connection between decreased hippocampal neurogenesis and depression. In a subsequent paper, Santarelli et al. (2003)  demonstrated that the behavioural effects of antidepressants in rats did not occur when neurogenesis was prevented with x-irradiation techniques. Very recent papers have linked together learning and memory with depression, and have suggested that neurogenesis may promote neuroplasticity. For instance, Castren (2005)  has proposed that our mood may be regulated, at a base level, by plasticity, and thus not chemistry; for instance, the effects of antidepressant treatment are only secondary to this.Caveats: The connections between mood and plasticity, and plasticity and neurogenesis, are new, and thus still under debate. Also, demonstrating even a strong link between neurogenesis, plasticity, and depression is a far cry from creating a general 'network theory of mood', as Castren is certainly aware. It's clear that there's much more to mood than just neurogenesis. But, frankly, the hypothesis of the past twenty-odd years that baseline mood arises from specific concentrations of certain chemical signals in the brain seems rather unpredictive and tapped out, whereas the hypothesis that mood depends on the communication patterns of one's neurons seems quite a generative model.
For what it's worth, the most common ways to actively promote neurogenesis that I've seen quoted in the literature are antidepressants (though they're a bit of a scattershot solution), exercise, learning new things, and being in love.
6-28-06: Reworked for clarity and accuracy.