There are three major sources of genetic diversity within a population. Gene flow requires the participation of a second population; it is based on new genes coming into the population via immigration of new gametes. In animals with internal fertilization, this must be accomplished by migration of animals. In animals with external fertilization, either individual animals can migrate, or their gametes alone can move into a new area, bringing new genes to the population they arrive in. For plants, pollen or seeds can be blown or carried to new areas by a huge variety of dispersers.
The second source of genetic diversity is mutation. Random mutations can be caused by a number of factors; wear and tear, radiation such as cosmic rays or ultraviolet light, and a lengthy list of chemicals all can cause mutation. Many of these mutations are deleterious to the individual, but some are passed into the population. Evolution is believed to be dependent on the accumulation of numbers of such mutations over long periods of time.
The third contributor to genetic diversity is procreation. Specifically, it's the crossing-over of chromosomes that occurs when cells undergo meiosis to produce gametes. While this does not actually bring any new genes into existence, it does shuffle the existing genes and produce new combinations, which can work differently in the individual who inherits them.
Another fascinating new science is epigenetics, which is the study of the non-gene parts of inheritance. The protein structure of chromosomes contributes heavily to the action of the genes within, and this effect, which is highly responsive to environmental influence, is another source of variation with a population.
There are several causes for genetic variation. Simple DNA mutations happen often. There is also genetic drift, which means that it's just a random change due to the sample population.
Some other causes that aren't as common would be Founder's Effect, usually caused by inbreeding of a small gene pool, and Bottleneck Effect is similar but usually caused by a rapid decrease in population.
Of course, Natural Selection is working at all times and often determines which genes will continue to be passed down and which will be bred out of the population.