Stars and planets are traditionally differentiated based on two properties:
(i) Whether or not they undergo nuclear reactions that burn hydrogen in their cores. Stars do this; planets don't. In order to have high enough temperatures in the core to burn hydrogen, an object needs to have a mass of at least 75 or so times that of Jupiter. Anything more massive than that is automatically considered a star.
(ii) The way they form. Stars form when a cloud of gas, out in a nebula or other region of interstellar space, collapses under the influence of gravity. Planets, on the other hand, form when material in the disk around a pre-existing star begins to condense around rock/ice cores. You can have situations where the entire planet is almost completely rock/ice/water (such as the Earth), or situations where a large amount of gas is subsequently attracted to the rock/ice core (such as Jupiter, Saturn, etc.).
There is actually some ambiguity in the above definitions, mainly because of the existence of objects called "brown dwarfs". Brown dwarfs are too small to burn hydrogen, so they can't be considered stars, but most of them seem to form in the same way that stars do, often out on their own in a cloud of interstellar gas, so they can't really be considered planets either. The question then becomes, where is the boundary between a planet and a brown dwarf? What if you have an object that is, say, thirty times the mass of Jupiter but is located near a star? Is it a planet or a brown dwarf? Astronomers don't generally know the formation mechanism in that case, whether the object formed along with the star from condensing gas or whether it has a rock/ice core at its center like a planet.
Because of this problem, a lot of people in recent years have advocated a new, simpler distinction between planets, brown dwarfs and stars which doesn't include the formation process in it. Under this scenario, the boundary between brown dwarfs and stars is still around 75 times the mass of Jupiter, as above, but the boundary between brown dwarfs and planets is set at around 13 times the mass of Jupiter, since that is the mass at which objects reach high enough central temperatures to burn deuterium (an isotope of hydrogen which undergoes nuclear burning at lower temperatures than regular hydrogen does).
This page updated on July 18, 2015.