How does evolution occur by natural selection?

Over time many theories have been put forward by the scientific community about how the process of evolution occurs. However the most commonly accepted theory amongst scientists is that evolution occurs by the process of natural selection. This was put forward by Charles Darwin. He proposed that:

There is naturally occurring variation in any given population - this is due to the variety in alleles which give rise to a variety of phenotypes (observable characteristics).

Those individuals who are better adapted for their environment as a result of the alleles they posess are more likely to survive and reproduce successfully.

Hence their offspring will inherit these alleles and so will also be more likely than others in the population to survive and reproduce etc due to the inheritance of these advantageous traits. 

Over a long period of time as the process of natural selection continues, there is a gradual change in the characteristics of a population by which members become better adapted for the environment in which they live. This is evolution by natural selection.