Super-powers are actually byproducts of neomorph mutations in human genetics, activated by a solar eclipse and triggered by adrenaline rushes. The mutations are a natural result of continuing human evolution, and most of them are benign or useless. Some few react in just the right way to create an extraordinary ability.
This ability is useless without the unique solar radiation generated by a solar eclipse. If an organism possesses this rare mutation, and is exposed to the eclipse radiation, the organism will manifest super-powers. The specific ability that manifests depends on two factors: genetic predisposition (Arthur Petrelli gains power from others, so Peter Petrelli gains power from others) and situational adaptation (Nikki isn't strong enough, so her dissociative personality Jessica manifests super-strength).
In reality, there are only a handful of super-power concepts out there, and most are just specializations of larger ideas (Ted Sprague can generate an EMP because EMP generation is a specialization of radiogenesis). The mutation causing electromagnetic pulse generation is nearly identical to the mutation causing radiogenesis, thus their similarity. The theory of power-families is based on that similarity.
DNA Alternates have manifested multiple powers, but they are always within the same power-family. Hiro Nakamura can travel through time and space because the mutation causing chronokinesis and the mutation causing instantaneous transmission are both specializations of the mutation causing gyrokinesis.
The mutations are similar in power-families, and the difference lies in the unique way an Alternate's adrenaline reacts with that ability, providing the specialization. That's why Alternates with the mutation causing intuitive aptitude can study the pituitary and pineal glands to understand the specialized adrenaline-ability interaction and replicate the process to copy a power.
Many abilities have multiple, compounding mutations. Take Meredith Gordon's pyrokinesis for example. Technically, she uses molecular manipulation to create combustion and energy manipulation to control the thermal energy. Because of these subtle connections, a DNA Alternate could feasibly have a molecular ability and pyrokinesis.
Other terms you may hear regarding such powers include the 'JoaT Regulation' and 'Advanced' powers. The Jack of All Trades Regulation is simply that all powers are equal in strength, but when a DNA Alternate has more than one power, that strength is divided between them. DNA Alternates with two powers have a primary and a secondary from the same power-family, with the secondary being weaker than the primary. The same applies to DNA Alternates with powers that allow them to replicate or take powers from another, termed 'Power-Farmers.' Read more about the JoaT Regulation here. An Advanced power is such a power that accesses multiple specializations of the mutation responsible for the Advanced power (Ted Sprague is an Advanced radiogenetic who can access the EMP generation specialization). The JoaT Regulation also applies to Advanced powers.
A unique instance of super-powers is the Invidia project. Created by a DNA Alternate with superhuman intelligence and perfected by a DNA Alternate with ingenious design, Invidia is a toxin lethal to ninety three percent of its subjects, but grants above-average super-powers with wild instabilities in the remaining seven percent. Invidia survivors have greater base strength than normal DNA Alternates, with less limitations. All Invidia survivors suffer from the unstable Invidia factors, which can range from heightened emotional influence to outright uncontrollable destruction.