A meteor is a piece of space material which enters the earth's atmosphere, and burns up in the atmosphere, leaving a bright track in the sky. Or, it is that track itself. This track typically lasts for a second or so, but can be more persistent.
A comet is an object, generally from the outer solar system (the Oort Cloud), which includes volatile material, such as ice. If and when the comet's orbit is disturbed so that it visits the inner solar system, these volatile materials boil off, carrying with them much of the material of the comet. The ejection of this material causes the comet to form its characteristic 'tail' of dust and gas. Unlike a meteor's track, this tail can last for thousands of years or more as the volatile materials slowly evaporate.