Your natural tendency to pick one hand or the other for a dominant hand is determined genetically (this is the answer to almost every nature v nurture question). Here’s an example recent article:
We’ll eventually get rid of questions that wonder if a particular trait has a mostly genetic underpinning; some will just take longer than others to identify the genes involved.
Anyway, the genetic influence here tends to make you right or left-brained (in simple terms), and of course since most of us have dominant left hemispheres for motor control, about 90% of us are right-handed.
Now that doesn’t mean, as has been pointed out, that you can’t train the other side to perform well–even equally well–for a given motor function. And of course if you lose part of your motor cortex early enough in life, the brain can do amazing things taking over for the lost part. So there is a substantial opportunity for nurture to make what would otherwise be the clumsier side more facile.
There are other ways besides handedness to decide which is your dominant hemisphere (this is in itself a simplistic concept). You can point to something and see which eye was dominant, for instance, by closing each eye one at a time. Or make a circle with your index finger and thumb, hold out a hand and enclose a distant object, and see which eye is being used. Once you realize what you are doing, though, these tricks become conscious awareness and don’t work as well.
And as pointed out re Phil Mickelson, or with computer mousing, it can be simply how you were nurtured if it’s a particular motor behaviour. I shoot a gun righty, a bow and arrow left, write right-handed and have a left dominant eye. The Pedant is a barely functioning kludge, alas. Damn genes.