The smallest "stars" are called "brown dwarfs," which are heavier than about 13 times the mass of Jupiter. Early in their lives, brown dwarfs do a little bit of nuclear fusion, burning deuterium (hydrogen with an extra proton) to make helium. But brown dwarfs are too wimpy to keep up nuclear fusion, and once all the deuterium is gone, the slowly cool off and fade away. Stars like the sun, on the other hand, get hot enough in their centers to continue nuclear fusion by burning normal hydrogen into helium steadily for millions, even billions of years.
For things smaller than 13 times the mass of Jupiter, no nuclear fusion will ever happen. For this reason, some astronomers think anything this small or smaller should be called a planet.
However, other people disagree. There are two ways to form objects this small. One is to form them like we think the planets of our solar system formed -- rocks and gas in a disk surrounding the parent star slowly build up to make planets. Some, like the Earth, would be pure rock, while others, like Jupiter, would have a rocky core with a very thick, massive atmosphere on top. Things that form this way would definitely be called "planets" by most every astronomer.
The other way to form giant planets is similar to the way stars form -- a large cloud of gas breaks up and shrinks under gravity. This could happen in isolation, or in orbit around another star (this is how multiple star systems are formed). "Planets" formed this way do not have rocks at their center, but are just large clouds of gas. They would not even need to be in orbit around another star, but floating freely in space. So, are these "planets?" I think most people would say "no," but astronomers are split on the issue.
The problem is that it is not possible to tell in which way a massive object formed. A 5-Jupiter-mass object detected around a star could have formed by either way, but it is virtually, if not fully, impossible to tell how it formed. For this reason, most atronomers lean toward a definition based purely on a planet's mass, not how it was formed.
I read this morning that the Internation Astronomical Union (IAU), which by agreement among astronomers has the power to define and to name all objects in space, is finally going to decide on all of these issues. The IAU has been talking about this for years, but the discovery by Mike Brown and collaborators is finally pushing them to come to a decision. It seems to be typical politics that now a decision has to be rushed, especially since astronomers have suspected for years that the Kuiper Belt had objects larger than Pluto, and since many objects more massive than Jupiter have been found around other stars and floating free in space.
The definition of a planet is likely to be an object that is too small to have ever undergone nuclear fusion, is large enough that gravity makes it round (instead of the potato shape of asteroids and comets), and that orbits another star. While I personally would like to include the way an objects is formed, I seriously doubt that can be part of a viable definition. With this definition, though, there would suddenly be many more "planets" in our own solar system, including a a handful of large asteroids and several objects in the Kuiper Belt -- as many as 23! Astronomers may just have to wait for pop culture to decide what a planet is.