Cell division is limited by a process called cellular senescence, during which cells in culture divide more slowly before stopping entirely. The cells do not necessarily die during senescence, but they no longer replicate.
The overall limit on cell division is called the Hayflick limit after the researcher who discovered it. Cells in culture tend to only divide a certain number of times before they become senescent. The causes of this senescence are still researched today, but some possible causes include telomere shortening and ongoing oxidative stress. Telomeres are non-coding sequences at the end of vertebrate chromosomes that get progressively shorter with each cell division. Oxidative stress is caused by reactive oxygen species over time. It is possible that this cumulative damage contributes to the Hayflick limit.