A couple of years ago there was a big brouhaha over whether Pluto is a planet. The International Astronomical Union declared that it is not. The controversy has popped up once again, as seen on blogs such as Bad Astronomy, Universe Today and The Planetary Society. The problem arose because when Pluto was discovered in 1930, it was believed to be Planet X, a large planet which was perturbing the orbit of Neptune. Well, better observations proved that there is no such perturbation, and that Pluto is relatively tiny, smaller than our moon. It was unlike all other planets in other ways, as well. Then astronomers started finding similar objects, some of them nearly as large as Pluto, and eventually they found one which was larger. Astronomers such as Neil DeGrasse Tyson had already been disputing Pluto's classification as a planet, and this brought the controversy to a head. Either we had to accept this new object, and possibly many others, as planets, or admit that Pluto is not one.
The problem was that there was no accepted definition of what a planet is. Originally, the Greeks called anything that moved in the sky a planet, which included the star-like Mercury, Venus, Mars, Jupiter and Saturn, as well as the sun and moon, but not the earth. After Copernicus and Galileo showed that most of these orbited the sun, rather than us, the earth was recognized as being a planet, and the sun and moon were recategorized. This worked fine, until a couple of hundred years ago when astronomers accidentally discovered Uranus orbiting far past Saturn, which fit in with the other planets. They wondered why there was a large gap between Mars and Jupiter, and when they looked found something, a small, round body which was recognized as a planet and named Ceres. However, Ceres was tiny compared to other planets, and then other small bodies were found in the same region, which were then reclassified together as "minor planets", or asteroids. Around this time, oddities in the orbit of Uranus led to the discovery of Neptune, making for eight planets.
Inaccurate measurements of Neptune's orbit led to the belief that there must be another large planet, called Planet X. This led to the discovery of Pluto by Clyde Tombaugh in 1930. As with Ceres, after several decades and the discovery of numerous other bodies in the same region, Pluto was demoted. Some people are still unhappy about this, and consider Pluto to be a planet. This all begs the question of what is a planet? It really comes down to semantics. It's easy to classify things when the differences between them are large, but when there is a range of intermediate objects, then the question of where to draw the line becomes rather arbitrary.
The most obvious place to draw the line is on whether the object is round. This would include little Ceres as well as Pluto and a number of objects past Neptune. The problem with this is that there may be a great many small, round objects. (That a planet must be orbiting the sun, and not some other body in the solar system, is agreed by everyone; then it would be a moon.) There are also some smaller bodies that are kind of round, and no planet is perfectly round, so how round does it have to be? It is even possible for a non-round object to be larger than a round one. And then there is the question of planets outside our solar system; some are large enough that they border on being small, dark stars. And obviously none of these are orbiting the sun; there are probably planets out there which don't orbit stars, having been kicked out by some other object at some point.
Two years ago, the IAU declared a new definition for a planet, which had three criteria: it has to orbit the sun, be round, and also contain most of the mass in its region. Ceres and Pluto do not qualify on the last count, and so they were designated "dwarf planets". Now, they want to create a new category, called "plutoid".
My opinion is, that it does not matter whether we call Pluto a planet or not. It does not change what Pluto actually is. It's simply a matter of convenience, not a real distinction. I'm inclined to not include Pluto, rather than adding a lot of similarly sized objects, but it really doesn't matter. It's just as important that it be studied, no matter what we call it. And fortunately, there is a space probe already on the way there.