Handedness was an individual's preference for using one hand over the other. The favored hand was usually referred to as the dominant hand. Due to being much more in use throughout one's life, the dominant hand was usually notably stronger and more dexterous than the other hand. Handheld weapons such as phasers were made to accommodate the user's handedness.
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")