Questions about apoptosis
I think this question could basically be summarized as "What is apoptosis?"  I will attempt to give a brief answer to that question, answering the two aspects of the question along the way.

Apoptosis is at its root programmed cell death (PCD).  It is basically when a cell (healthy or not) dies so that it can be replaced.  It is happening all the time in multicellular organisms such as me, you, and the lion in the zoo.  Basically, it allows all the cells to be replaced as the tissues of the body grow.

For instance, as your skin grows, cells in the dermis are dividing to produce a new epidermal layer.  At the same time, cells on the outside are dying (at the stratum corneum) so that the skin does not grow infinitely and we aren't covered with 20 inches thick of skin by the ends of our lives.

The same thing is happening in, for instance, the liver.  As the liver cells split, old ones are also dying.  This allows us to rebuild the liver to its normal size after a surgery in which we remove a section of it.  We simply have to stimulate cell growth so that it counterbalances PCD (apoptosis) and the liver grows to its normal size.

Cell death is initiated, not by the DNA, but rather by caspase activity.  All cells have inactive procaspases which will destroy the cell as soon as they become active.  When what we call an initiator procaspase bonds with a procaspase within the cell, the procaspase is activated, and the signal is spread across the cell.  This initiator procaspase is usually triggered by a death receptor outside the cell.

More reading here