Why do I have darker, dry skin on my hand? It's only in certain areas and only on my right hand.

Moisturize. There are many reasons for patches of dry and dark skin on your hands - it could be just dry skin that is darkened by the sun, it could be atopic dermatitis (which is just irritated skin from being so dry), it could even be a mild fungal irritation. First thing to try is gentle moisturizers and sunscreen!