(covers information from several alternate timelines)
Hunting was the practice of stalking and sometimes killing animals, and occasionally sentient species, either for meat, sport, or for other resources, such as fur or feathers which could be used for clothing.
According to Captain Jonathan Archer, hunting went out of style on Earth sometime before the mid-21st century. However, it was still practiced by the Humans inhabiting Alcor IV in the 33rd century. (ENT: "Rogue Planet"; ST: "Calypso")
The rogue planet of Dakala was home to a species of telepathic shapeshifters who were hunted for sport by the Eska. It also contained the drayjin, pig-like creatures which were hunted for their meat and for sport as game animals. (ENT: "Rogue Planet")
Members of the Tosk species were specifically bred as quarry for another race from the Gamma Quadrant, known as the Hunters. The very name of the species itself meant "The Hunted". Everything about the species was designed to make them elusive prey and the rules constructed by the Hunters allowed them to fight back and even kill their pursuers. They would use any means necessary to outwit the Hunters, but considered it a great shame to be captured alive, as they would be placed on display and made fun of. (DS9: "Captive Pursuit")
On Gaia in an alternate timeline, members of a group known as the Sons of Mogh, which practiced the Klingon way, routinely rode wild torgas and hunted within the planet's wilds, visiting the settlements only to trade furs for things they needed. (DS9: "Children of Time")
Some members of Chakotay's tribe hunted deer. This, however, was something which Chakotay himself refused to do. Thus, he found it odd that he had a dream in which he hunted a deer with a spear, which morphed into a member of the dream species. (VOY: "Waking Moments")
The Hirogen hunted sentient species throughout the Delta Quadrant. The entire culture of their species was based around the hunt, but some Hirogen recognized a need to change, lest they hunt themselves to extinction. (VOY: "The Killing Game")