dominant hand
Refers to the hand one naturally prefers for writing, throwing, etc. Common in everyday conversation.
My dominant hand is my right hand.
skilled person; person of ability
Rare, somewhat old-fashioned usage. Refers to a person who is highly skilled or an expert, often in a craft or art. More commonly expressed as 腕利き or 名人 in modern Japanese.
He is a master-hand in this field.
I injured my dominant hand and practiced writing with my left hand.
Compound of 利き (kiki, 'effective, working') + 手 (te, 'hand'). The 'skilled person' sense is an extension of the idea of a 'working hand'.