“Although the percentage varies by culture, in Western countries 85 to 90 percent of people are right-handed and 10 to 15 percent of people are left-handed. Mixed-handedness (preferring different hands for different tasks) and ambidextrousness (the ability to perform tasks equally well with either hand) are uncommon.”
Current theory suggests that multiple genes contribute to the development of handedness - but studies with twins have shown that only around 25% of handedness is hereditary.
There are various theories regarding why handedness may (or may not) be beneficial, and why it may have established itself as such a persistent trait (in humans).
There is currently no explanation as to why right handedness has become dominant over left handedness.
Further reading: Is handedness determined by genetics? US National Library of Medicine
Also see; Wikipedia