Regeneration, in the biology context, is the ability to regrow damaged tissue. Known as a heal factor in the science-fiction genre, it is arguably the most useful ability to have and could be the key to immortality.
Most animals regenerate, with the exception of mammals. Mammals, when injured, will get scarring. They also grow old. Both of these traits originate from Gene P21, which mammals evolved during the time of the dinosaurs as a population control and to reduce competition for food. Gene P21 inhibits our natural regenerative powers and suppressing it in laboratory mice has been proven to restore the ability.
Regenerating has some interesting effects on the individual. Because DNA is damaged slightly every time the cell divides, fast healers should be more prone to cancer, as their cells divide more often. Fortunately, our immune systems are programmed to kill damaged cells, creating a shorter cell life cycle for the person who regenerates. Of course, this means we have to regenerate those damaged cells. Whether or not this escalates is unknown, although it doesn't seem to have much effect on non-mammals. Seriously, tortoises can live for 200 years without getting cancer.
Healing fast, as the perceptive amongst you will have figured, creates a faster metabolism and thus requires more food. If you plan on getting healing powers, you'd better be prepared for a massive grocery bill. That's one thing they never show us on Heroes - Claire Bennet should be having to eat constantly.
I said regeneration was the key to immortality. That's because your cells are constantly replacing themselves, so your body never wears out. Why do you think tortoises live to 200? Reptiles keep living until something else kills them, while mammals are weakened by the P21 gene and eventually their hearts give out.
As of yet, there are no reported cases of humans regenerating. The consequences immortality would have upon society are massive, as you can probably imagine.