Two typical comets that I know are Donati's and Halley's. So comets also have orbits, but are their orbits on the same plane (such as the ecliptic) as the other planets?
Those two comets happen to have large orbital inclinations to the Ecliptic: 117$^\circ$ for C/1858 L1 (Donati) and 162$^\circ$ for 1P/Halley. Long-period comet orbits are oriented essentially at random, suggesting that they originate in the Oort cloud, a sphere several thousand AU from the Sun. Many short-period comet orbits are inclined only a few degrees to the Ecliptic, suggesting origins in the scattered disc, a torus several tens to a few hundred AU from the Sun.