Some cells have very short lifespans because their jobs are completed quickly, (disease-fighting cells in the human blood stream are an example) but others just seem to "wear out." A part of this "wearing out" results from MITOSIS.

Each time human DNA divides, it degrades. Human cells can only divide (go through mitosis) about 50 times. This is called the "Hayflick Limit." Eventually the DNA is no longer able to divide, so the cell dies, so the tissue cannot repair itself and dies, so the organism dies.

Here are a few links that go into greater depth

So how does the cell itself die, after it is no longer able to divide?
  1. It blows up and rots (necrosis)
  2. It kills itself! This is called apoptosis: programmed cell death.
    Here are some links to apoptosis