Fundamentally, nerve cells (neurons) aren't terribly different from any other cell type. They function according to the same principles, have nuclei, DNA, cytoplasm, etc. and are therefore subject to the same vulnerabilities as most other cells, such as infection, starvation, mutations, programmed cell death, as well as mistakes or hijackings in any of the standard cellular processes such as apoptosis.
A common claim is that organisms are given a fixed number of neurons sometime between birth and early adulthood, and that these neurons only diminish over time. Following this pattern we would therefore expect to see a trend of diminishing mental and physical capacity over an organism's lifetime. The reality is a little more complicated than this, but in general we expect to see cumulative neuron death, or neurodegeneracy, over a lifetime. As for why, again there is no unique reasons why neurons die as opposed to any other cell type; the same mechanisms apply to all. We can generalize by saying that the body attempts to maintain its necessary functions while minimizing risks due to aging and malfunctioning cells, weeding out the cells posing the highest risk.
Neurons are a little more unique in being subject to the effects of excitotoxicity, which is basically just a way in which their normal transmitting functions are abused and result in cell death. This is one way in which drug use can kill neurons. Many neurotransmitters function via the use of a particular molecule bonding to a site and triggering an electrochemical signal, often by releasing ions. Excess ion buildup in turn triggers other cell processes, and an overwhelming number of ions can simply tell the cell to kill itself, whether intentionally or not.