To understand what a long-term comet is one needs to understand the differences between long-term and short-term comets and how they differ from other celestial bodies like asteroids. Asteroids are small stone or iron bodies generally found inside the orbit of Jupiter. They contain little or no volatile ices.
Comets are small celestial bodies with significant volatile ice content and orbits that approach the sun. They get the name comet from the evaporation and ejection from the comet nucleus of water and other volatile gases, along with dust, when these bodies are heated as they approach the sun. These gases and ejected dust form a tail or coma as they are pushed away from the cometary nucleus by solar winds. The tail or coma is what visually separates comets from asteroids, although there are some comets that have lost all their volatile ices after repeated cycles near the sun and no longer display a tail or coma.
Comets can be divided into two general classifications, short-term or short period comets which have orbits of less than 200 years in duration and long-term which have greater than 200 year cycles.
Short-term comets have elliptical orbits and return to perihelion, or closest approach to the sun, on a regular predictable basis. They are also called periodic comets by the International Astronomical Union Minor Planet Center (MPC). The well known comet Halley is a short-term or periodic comet and carries the designation P/1682, P/ for periodic and 1682 for the year it was first observed. Periodic comets are thought to have originated in the Kuiper belt, a disk shaped region beyond the orbit of Neptune consisting of mainly small icy bodies similar in composition to comets The dwarf planet Pluto is a Kuiper belt object. Periodic comets could also be long-term comets that have been captured in shorter term orbits. Periodic comets tend to orbit near the same plane as the planets.
Long-term comets have orbits longer than 200 years. Some may have orbits that last millions of years. A rare few called hyperbolic comets make a single pass around the sun and then travel out of the solar system entirely. The orbits of long-term comets tend to be more eccentric and less predictable. They spend more time at aphelion and at slower speeds and can be affected by the gravity fields of the larger outer planets. Long-term comets are thought to originate in the Oort cloud, a spherical area a thousand times more distant from the sun than the Kuiper Belt. The Oort cloud has not been observed, but theorized because long-term comets come into the inner solar system from all directions, not just the planetary plain, at velocities that indicate origins in this spherical area. The Oort cloud objects are so far from the sun that gravity fields of other stars or the Milky Way Galaxy itself may perturb the objects and start them falling into the inner solar system. The influence of extra solar gravity effects and the possibility of collisions with other Oort cloud objects could help explain some eccentricities of long term comet orbits.
We still have much to learn about the Oort cloud and long-term comets. New telescopes, observation techniques and ways to interpret data will keep adding to our knowledge of comets and their origins.
A list of known periodic comets.
Animation of outer planets and known Kuiper belt objects.