In the 160 years in which "handedness" has been studied we have learned quite a lot, but we still cannot precisely describe what causes humans preferentially to use one hand over the other, or why human populations are biased toward right-hand use rather than left-hand use.
Scientists disagree over what percentage of human populations are "right-handed" or "left-handed" because there is no standard, empirical definition for measuring "handedness"; our criteria vary, and are based on various theoretical explanations because we are still trying to understand the mechanisms involved. But I can describe in general terms what we do know.
Most humans (say 70 percent to 95 percent) are right-handed, a minority (say 5 percent to 30 percent) are left-handed, and an indeterminate number of people are probably best described as ambidextrous. This appears to be universally true for all human populations anywhere in the world. There is evidence for genetic influence for handedness; however, it is non-Mendelian and geneticists cannot agree on the exact process. There is evidence that handedness can be influenced (and changed) by social and cultural mechanisms. For instance, teachers have been known to force children to switch from using their left hand to using their right hand for writing. Also, some more restrictive societies show less left-handedness in their populations than other more permissive societies.
Some researchers argue there is evidence for cases of "pathological" left-handedness related to brain trauma during birth. And many researchers trace the cause of handedness back to pre-natal, interuterine developmental processes, back to the time when the fetal brain is first developing distinct cerebral hemispheres. In the 1860s the French surgeon Paul Broca noted a relationship between right-handedness and left-hemispheric brain specialization for language abilities. But the hand-brain association is neither a simple, nor reliable, correlation. Studies conducted in the 1970s showed that most left-handers have the same left-hemispheric brain specialization for language typical of all humans--only a portion of left-handers have different patterns of language specialization.
So the bottom line is, we have a good general idea of the causes of right-handedness in human populations, but we have yet to work out the precise details, including why the direction is right instead of left.
Do other primates show a similar tendency to favor one hand over the other?
The second question (do non-human primates show handedness) is currently a controversial one. It is important to note the difference between an individual animal being left- or right-handed, and most of the animals in an entire population being either left- or right-handed. It is not unusual for individual animals to show a preferential use of one hand over the other, to develop an individual hand preference. But there is no consensus among researchers that any non-human species shows the same species-level handedness found in humans.
There are a few researchers who argue for this, but most of these work with animals in laboratory or captive settings, performing manual tasks that are very different from how animals use their hands in the wild.
In addition to studying handedness in humans, I have also studied hand usage in mountain gorillas (in Rwanda) as well as chimpanzees, red colobus monkeys, redtail monkeys and grey-cheeked mangabeys (in Uganda). My own research shows that individual monkeys and apes often develop individual preferences (both left and right) for manual tasks, but I have found no evidence for population-level handedness, as seen in humans.