Usually it's damage to the pathway that activates this cell suicide. The body signals for it to die off but it doesn't respond to the signal. This can happen in numerous different ways, as evidenced by the multitude of types of cancers.

Edit: as an example, damage from ultraviolet light causes many cells to die off, in the form of sunburn and the blistering that follows. Some cells get mutated but don't die, and can become skin cancer.