Telomere attrition:

Telomeres are the region at the end of each chromosome in the nuclear DNA. When DNA replicates, it can’t quite replicate the very end of the chromosome. So the telomere is a repeated section of nucleotides (TTAGGG – repeated over a thousand times) at the end of the chromosome. Some of it gets left off each time DNA replication occurs. Eventually, when the cell has replicated a number of times, the telomere is short enough to trigger a sequence of events leading to either cellular senescence or cell death.

It isn’t quite as cut and dry as just hitting a limit on the number of times a cell can replicate. There is an enzyme called telomerase that can repair telomeres. It can add the repeats back to the end of DNA, restoring the length of the telomere. Telomerase is active in stem cells – and in cancer cells, which allows them to grow and divide without limits.