Classifying comets can be done in several different methods, including orbital period, orbital body, and orbital shape. The reason for such variation is that how comets orbit, where they orbit, and the orbiting body makes all the difference other than size of the comet itself.
Comets are nothing more than “cosmic snowballs of frozen gases, rock and dust, which when frozen, they are the size of a small town.”
If an object is called a comet, then the object orbits the sun. However, there is an object called an exocomet, which is a comet outside the Solar System that includes rogue comets and comets that orbit stars other than the Sun. The first exocomet was discovered in 1987, with a total of 27 stars having been observed or suspected of containing exocomets.
One of the simplest ways of categorising comets is by their orbital period. In their most basic categorisation, they come in two flavours. Short period comets and long period comets.
Short period comets, which are called periodic comets are generally defined as those having orbital periods of less than 200 years. They usually orbit more-or-less in the ecliptic plane in the same direction as the planets, which means that their orbits typically take them out to the region of the outer planets. 
Long period comets are comets that have highly eccentric orbits and periods ranging from 200 years to thousands of years. An eccentricity greater than 1 when near perihelion does not necessarily mean that a comet will leave the Solar System. This is important as the length of time that a comet rotates round the sun is crucial to determining the role it plays in the solar system that it lives in.