Apoptosis is a biological process which is often described as the process of "programmed cell death". The term is derived from the term from the medical term ptosis (n. Greek), meaning "weakening", "drooping", or "falling" (see Wikipedia). The prefix "apo-" in biology is often used to refer to unutilized potential (as in apoferritin meaning the ferritin protein which is not yet loaded with iron). Together, the terms would mean poised or ready for death, but not yet dead.
As cells are stressed or damaged, their biological processes are altered to stop cell division and repair the problems caused by whatever insult to which the cell was subjected. If it can recover, it resumes its life as a normal (albeit damaged) cell. Sometimes the damage is too great and the response to repair the cell is not enough. That the cell has ben damaged beyond repair is not sufficient to push the cell into apoptosis. Sometimes seerly injured cells do not die appropriately, they continue to divide inappropriately and may develop into problematic tissues- potentially even cancer.
If harmed cells do respond properly and proceed into apoptosis, the cell will begin to digest itself. There are proteases, nucleases, and general oxidative species which are released into the cell upon commitment to the apoptotic program. These spcecies are released from the mitochondrion in order to fragment proteins, DNA, and lipids within the cell to destroy its organelles, thus preventing it from continuing to divide. Fragmentation of DNA may help prevent transfer of genetic mutations to subsequent generations of daughter cells. Additionally, protease activity may potentially "quench" signals which the apoptotic cell may send to its neighbors. Lastly, degradation of the organelles serves as a general mechanism for the destruction of the cell, most likely by lipid oxidation (via cytochrome c).
Apoptosis serves as a mechanism to silence and ultimately kill a severly damaged cells. Commitment to this "suicide" program is not lightly undertaken and there is the possibility to "stand down" from this final judgement. This process is a necessity to preserve the integrity of the genome of daughter cells as well as the appropriate signaling between cells in a multicellular organism. As such, unicellular organisms are currently thought to be self-serving and do not engage in this behavior.