Neurogenesis simply stated is the birth of new neurons. It occurs during embryonic development when the nervous system is first formed from a single cell tube to a multilayered structure. Recent evidence has found that neurogenesis continues into adulthood in mammals and primates. Evidence is pretty strong for neurogenesis in two brain areas, the hippocampus (spatial memory) and the olfactory (smell) and has been postulated in other areas like the neocortex. Even in humans some evidence points to neurogenesis over a lifetime in these brain areas. So we are producing new neurons during our whole life. But at this time it is not clear what the function of these new neuron are or how long they even stick around. This question is the focus of much of the current research.
A cake-based model if embryonic neurogenesis from a class on Art and Neuroscience