In 2006 there was quite a controversy when the International Astronomical Union decided that Pluto, long held as the ninth planet of the solar system, was to be demoted. It would now be known as a Dwarf Planet.
What is a Dwarf Planet?
In most respects Dwarf Planets have the same characteristics as all the other planets that you are familiar with. They are objects in orbit around the Sun that are massive enough that gravity has formed them into a spherical shape.
The primary different between Dwarf Planets and regular Planets, is that Planets are said to have "cleared their orbital path of debris".
This is a decidedly vague term, and is the primary source of all the controversy. But under closer examination it becomes clear what the spirit of the condition is to impart.
Take the case of pluto: it is actually one of many small bodies orbiting in the Kuiper belt region. At least a few of these objects are of similar size to Pluto. So if you are going to include one of them, Pluto, in the category of Planet, then you would need to include them all.
But beyond that even, you have to examine the formation of these objects. Pluto, like the other Dwarf Planets, are likely the result of Planets that never formed properly. Pluto started out life as the building block of a full fledged Planet. But Neptune's gravity likely caused the planet to become unstable, ripping it apart into lots of smaller objects.
These objects are now orbit in a similar region as Pluto. That is to say, Pluto is not alone in its orbit around the Sun, and since it has not the mass to pull the rest of that material together into a single object it is classified differently than the other worlds of our Solar System.
Personally, I agree that objects like Pluto should be classified separately from the other eight Planets. However, I don't much like the term Dwarf Planet; I think Planetary Remnant is more descriptive. It conveys the reality of Pluto's existence, that it is the largest bit of a planet that didn't quite form properly.
But they didn't ask me. So we are stuck with Dwarf Planet.
Are There Other Dwarf Planets, Besides Pluto, in our Solar System?
As of this writing there are five objects listed as Dwarf Planets in our solar system. They are: Ceres, Pluto, Haumea, Makemake, and Eris.
Eris was once believed to be larger than Pulto, which is what sparked the discussion of planet definitions in the first place, but was recently determined to be smaller than Pluto by a tiny amount.
Charon, officially considered a Moon of Pluto, is sometimes mentioned as a Dwarf Planet since it is of similar size to Pluto. This makes some sense because Charon is of similar size (though still noticeably smaller) to Pluto and therefore they both orbit a point between them, rather than Charon orbiting Pluto in the traditional planet-moon configuration.
However, since there is no strict definition of what constitutes a Moon, it is difficult to argue one way of the other. So for now, Charon is generally left out of the discussion of Dwarf Planets.