According to the International Astronomical Union (IAU) a planet must satisfy three criteria:
1. It is in orbit around the Sun, as opposed to a Moon, which orbits another planet.
2. It is massive enough to achieve hydrostatic equilibrium, having formed itself into a spherical shape.
3. Has "cleared the neighborhood" of its orbit of other comparable objects, and is the gravitationally-dominant body in this neighborhood.
Pluto meets the first two criteria, but not the third. Though it is round and orbits the Sun, Pluto's orbital neighborhood, the Kuiper Belt, is littered with hundreds of other objects.
If this third criterion seems a bit arbitrary, well, it is. There was some pretty contentious debate between the pro- and anti-Pluto factions at the 2006 IAU meetings that finally set the definition of a planet. The anti-Pluto faction won the debate, and the IAU essentially wrote the definition of a planet so as to deliberately exclude Pluto. It is still controversial - after all, the Earth's neighborhood contains dozens of potentially hazardous asteroids that cross our orbital path, so we haven't quite "cleared our neighborhood" yet either.
However, it needed to be done. The definition of "planet" came from the Greek word "wanderer," referring to the way planets wandered from night to night against the background stars. This was easy to define before telescopes, but in recent years astronomers have found "wanderers" in the Kuiper Belt that are larger than Pluto and also had their own moons.
We were looking at a solar system of fifteen planets and counting - Eris, Ceres, Haumea, Makemake, Sedna, and others. Even Pluto's own "moon" Charon was also a contender for planetary status, since it doesn't really orbit Pluto. The two both pivot around a common center of gravity.
Now, we have a nice easy system - eight classical planets and five dwarves - a number certain to increase as new discoveries are made.