The chances for genetic mutation occurs when the sex cells are being made. DNA is not always copied 100% accurately.
Most of the mutations occur on parts of the genes known as "junk DNA" and so produces no immediately obvious effect.
Others affect hair color, skin complexion, size and shape of genitals, muscular size and strength, and other less measurable factors such as musical talent, spatial awareness and so on. In many cases a mutation does no direct harm to an organism, which then survives to pass the change on.
The most obvious example of this is the cumulative effects of mutation on the hair and skin color of human descendants as they fanned out across the globe.
A red-headed celt is unlikely to mate with another red-headed celt and produce a tiny west african bushman. However, each race is descended from common stock so the changes must have built up over time.
At present, the various human types have not lived in isolation for long enough for cross fertility to become impossible. Now, with travel and intermarriage it is unlikely to happen on a planetary scale, but if ever we travel to the stars, it will be interesting if a human descendant on another planet in say 20 000 000 years will still be fertile with one from Mother Earth.
Mutation of the sex cells has now been shown to happen in response to conditions experienced at time of their formation.
In the case of females, this is at the stage in gestation where the supply of egg cells is being laid down in the ovaries of the fetus. So a famine, drought, period of plenty, a stressful pregnancy with its associated effects, will have an effect on the grandchild of the pregnant female.
In the case of males, it is during the early stages of puberty, where sperm production begins, and at other stages of life in response to prevailing conditions. Again the effect will be on the grandchildren of his parents.
This study was done very recently, but its potential is very exciting.
Other mutations occur among bacteria, when they exchange DNA, which is then copied with variants, leading to other types of bacteria.
Of course their is also mutation caused by radioctivity, and cosmic rays which are so negligible in the natural world as to be hardly worth considering.
Damage to DNA in adults generally results in cancers - melanoma being the most obvious example, but this is not really mutation in Dawkins sense.
HB
A topic of debate is whether Neandertals and modern humans were cross fertile. Both are descended from a common ancestor. There is sufficient proof that the two groups co-existed for a long time. If at first they were able to intermarry, this would account for the very few examples of cultural cross-fertilization.
One theory is that as cross-breeding became impossible the less able Neandertals were hunted to extinction as "unclean" or dangerous to the superstious people of the period. Or just an inferior to be wiped out.