The short answer: A planet is a large body that orbits a star, and dominates the vicinity of its orbit.
The long answer: Believe it or not, until 2006 there was no formal scientific definition of the word planet. Most of us have been taught that there are nine planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto. The asteroids were at one time also considered to be planets, but scientists soon realized that they were different - there were lots and lots of them, and they were all in the same general region of the solar system, sharing orbits between Mars and Jupiter. Pluto was not discovered until 1930, and it was soon heralded as the ninth planet. But technology improved, and scientists began to discover that there were lots of other bodies a lot like Pluto in that same region of the solar system (called the Kuiper Belt). It eventually became obvious that Pluto should never have been called a planet, but by the time that happened, Pluto was entrenched. Popular culture would not allow scientists to "demote" Pluto to non-planetary status. So scientists left it alone. After all, none of the other Kuiper Belt Objects (or KBO's) were as large as Pluto, so the problem could be ignored.
In 2005 an object named Eris was discovered in the Kuiper Belt beyond the orbit of Pluto. And it is larger than Pluto. The discovery of Eris forced the scientific community to come up with a formal definition of the word "planet," and that definition corrected the misclassification of Pluto as a planet.
