A comet is a celestial body, usually found orbiting a star, typically characterized by a solid nucleus of rock and frozen water and surrounded by a nebulous coma. Under increased exposure to heat energy as it approaches perihelion (an orbit's nearest point to a star), a comet's nucleus sheds mass away from the star, creating a sometimes spectacular "tail".
Notable comets
Sol's best known comet, Halley's Comet, has a regular orbit of approximately 76 years, and is often visible from Earth's surface when passing. Coincidentally, Halley's appearances marked the years of the birth (1835) and death (1910) of the famous American writer Samuel Clemens. In a time travel incident that brought him from San Francisco of 1893 to the decks of the USS Enterprise-D in 2369, Clemens was curious if the spaceship had visited Halley's Comet. (TNG: "Time's Arrow")
Archer's comet was discovered by the Earth starship Enterprise NX-01 and named for her Captain, Jonathan Archer, in June of 2151. With a diameter of 82.6 kilometers, the comet was the largest yet observed by Humans. More unusually, the nucleus contained large quantities of eisilium under a thick magnesite surface layer. (ENT: "Breaking the Ice")
Comet Burke
In the 22nd century, many comets in the Sol system were harvested for their resources by the Mars terraforming project. The Martian verteron array fired verteron pulses to divert the orbit of comets, starting years in advance and guiding their course with care, into a safe impact on the Martian north polar ice cap. At one point, fourteen comets were scheduled for Martian consumption in a thirty-month span, starting with the arrival of comet Burke in January of 2155. (ENT: "Terra Prime")
Comet Icarus IV in the Romulan Neutral Zone was the locus for a portion of the famous starship battle between the USS Enterprise and a Romulan Bird-of-Prey during the Neutral Zone Incursion of 2266. (TOS: "Balance of Terror") A century later the Enterprise E would detect a class 2 comet in the Romulan Neutral Zone while on patrol. (Star Trek: First Contact)
The Enterprise-D tracked a rogue comet through Sector 1156 in 2370. The core of the nucleus, composed largely of fortanium, proved to be a D'Arsay cultural archive, built over 78 million years ago. (TNG: "Masks")
A comet in the Bajor system threatened the Bajoran wormhole in 2371, and became known as the Sword of Stars. According to ancient Bajoran prophecies, the Sword of Stars would destroy the Celestial Temple. (DS9: "Destiny")
Background
In the Sol system, comets often originate from the Oort cloud, a collection of ice, dust, and other debris particles that marks the outer boundary of the solar system. Other comets are known to have come from the Kuiper belt.