Handedness was an individual's preference for using one hand over the other.
Geordi La Forge was right handed. (TNG: "Ship In A Bottle") Beckett Mariner was left handed. (LD: "I, Excretus") Ezri Dax was right handed before being joined with the Dax symbiont. (DS9: "Afterimage")
In 2365, Data, while playing Sherlock Holmes on the holodeck, asserted that a left footed person was also left handed. (TNG: "Elementary, Dear Data")
One's handedness could be determined by an analysis of one's handwriting. In 2369, Data, as Holmes again, attempted to prove that a right handed murder victim's left handed brother wrote a suicide note by throwing a box of matches to the man. The accused caught it with his right hand due to a holodeck malfunction. Later, when assessing the handedness issue, Reginald Barclay asked James Moriarty if he was right or left handed, to which Moriarty replied that he was left handed. (TNG: "Ship In A Bottle")
Some pitchers had a dominant hand. In 2371, Benjamin Sisko asked Kasidy Yates if the Pike City Pioneers could use a right handed pitcher. (DS9: "Family Business")
A boxer or fighter who was left hand dominant was called a southpaw. (VOY: "Tsunkatse")