Poor old Pluto is not longer a planet
So how does the definition of a dwarf planet differ from that of a regular planet? A dwarf planet is almost identical to a regular planet except for one thing, it hasn't cleared the neighborhood around its orbit (part "c"). And just so you don't get confused, if something that would otherwise be a dwarf planet is orbiting something besides the sun, it is not considered to be a dwarf planet.
Confirmed and likely dwarf planets
The question, that finally knocked Pluto from its position as a planet, of what exactly a planet is, was posed when we discovered a little celestial body in orbit beyond Neptune (known as a trans-Neptunian object, or TNO) known at the time as 2003 UB313 (Now called Eris). This object turned out to be bigger than Pluto, so someone had to decide if we were going to call it the 10th planet or not. It is the IAU that decides such things, so they convened to settle the matter. It was their decision that gave rise to the official definition of a dwarf planet and Pluto's fall from glory.
The International Astronomical Union is in charge of making big astronomy decisions
It turns out that there are a lot of objects orbiting our sun, and I don't just mean asteroids. So far we have found nearly 1,000 TNOs, of which a significant portion could be dwarf planets. 200 dwarf planets are expected to be found when we've explored the Kuiper Belt, and 2,000 more even farther out. Our solar system is filled with these less-than-planetlike objects.