Nerve cells don’t actually divide. What happens is, there is a population of cells called neuroblasts, neuro which means it is something to do with nerves and blast which means progenitor or stem cell like. These cells divide and their daughter cells go on to become nerve cells.
Once a nerve cell or as scientist call them neuron is a fully grown up neuron, they do not divide anymore.
But as every other mature cell type in our body, neurons itself have to become neurons first. Neurons are derived from a cell group called neural crest cells, and these cells can and do actually divide. Cell division follows tha same program in every cell. First the DNA has to be duplicated and then one cell can grow and be divided into two daughter cells which each containing one set of DNA.
Nerve cells are very important, as you know, we could not think, smell, see, hear or feel without them and some people when they get older can loose some nerve cells due to damage (accidents) or ageing (remember the retinal pigment epithelial cells and macular degeneration from the christmas lecture) . As nerves themselves can not divide anymore and therefore can not regenerate, most tissues containing neurons in our body will also hold some of these progenitor cells, stem cells which can develop into nerves. So once a damage has occured, these stem cell can divide and make new nerves, which can then replace the damaged neurons.