For thousands of years it has been known that planets and stars are two distinct classes of celestial object. The very word "planet" is derived from a Greek term of similar pronunciation meaning "wanderer." Why wanderer? Because the planets moved through the sky in a fashion wholly unlike the circular paths carved by stars - they wandered about. Aside from this superficial and admittedly earth-centric difference there are three physical characteristics that allow an inquisitive mind to discern between a star and a planet, and they will be arranged in order of increasing robustness.
3) Presence of naturally occurring, sustained fusion
Size is the first factor we will address that can be used to tell whether an object is a star or a planet is the size of the object. Stars tend to be very large when compared to all other gravitationally bound bodies in a given solar system (like planets, moons, and so forth). Unfortunately, this trait is really not particularly useful for several reasons: stars vary in size by many orders of magnitude, some systems have multiple stars within them, and dead stars shrink to planet size or smaller, to name a few.
Composition is another way to tell a star from a planet. Stars tend to have enormously high proportions of hydrogen (H) and helium (He) compared to the other elements present. Terrestrial planets, on the other hand, are composed primarily of much heavier elements. This trait becomes less useful when dealing with gas giants, however. Gas giant planets like Jupiter are sometimes called "failed stars" because of the high percentage of H present in such worlds. If one is to more precisely distinguish between a planet and a star, more factors are needed.
The most surefire method to determine if a given body is planet or star is to find out whether there are sustained nuclear fusion reactions occurring on the body. The gravity produced by an object as massive and, more importantly, enormously dense as a star squeezes it's component atoms together with such force that they actually fuse, creating heavier atoms. This activity is called nuclear fusion and, due to a strange quirk of quantum mechanics, releases enormous amounts of energy. This energy is then expelled from the star in the form of electromagnetic radiation.
So, the short answer to the question that inspired this piece is as follows:
If the celestial object in question is generating enormous amounts of electromagnetic radiation across a broad spectrum (in particular, radiation of a higher frequency than visible light such as UV radiation), it's a star. Planets reflect electromagnetic radiation and emit some infrared radiation, but that is all.