How many planets are there in this solar system?

In addition to Undo's fine answer, I would like to explain a bit about the motivation behind the definition.

When Eris was discovered, it turned out to be really, really similar to Pluto. This posed a bit of a quandary: should Eris be accepted as a new planet? Should it not? If not, then why keep Pluto? Most importantly, this pushed to the foreground the question

what, exactly, is a planet, anyway?

This had been ignored until then because everyone "knew" which bodies were planets and which ones were not. However, with the discovery of Eris, and the newly-realized potential of more such bodies turning up, this was no longer really an option, and some sort of hard definition had to be agreed upon.

The problem with coming up with a hard definition that decides what does make it to planethood and what doesn't is that nature very rarely presents us with clear, definite lines. Size, for example, is not a good discriminant, because solar system bodies come in a continuum of sizes from Jupiter down to meter-long asteroids. Where does one draw the line there? Any such size would be completely arbitrary.

There is, however, one characteristic that has a sharp distinction between some "planets" and some "non-planets", and it is the amount of other stuff in roughly the same orbit. This is still slightly arbitrary, because it's hard to put in numbers exactly what "roughly" means in this context, but it's more or less unambiguous.

Consider, then a quantity called the "planetary discriminant" µ, equal to the ratio of the planet's mass to the total mass of other bodies that cross its orbital radius and have periods up to a factor of 10 longer or shorter. This is still a bit arbitrary (why 10?) but it's otherwise quite an objective quantity.

Now take this quantity and calculate it for the different bodies you might call planets:

Suddenly, a natural hard line emerges. There's a finite set of bodies that have "cleared their orbits", and some other bodies which are well, well behind in that respect. Note also that the vertical scale is logarithmic: Neptune's planetary discriminant is ~10,000 bigger than Ceres'.

This is the main reason that "clearing its orbital zone" was chosen as a criterion for planethood. It relies on a distinction that is actually there in the solar system, and very little on arbitrary human decisions. It's important to note that this criterion need not have worked: this parameter might also have come out as a continuum, with some bodies having emptier orbits and some others having slightly fuller ones, and no natural place to draw the line, in which case the definition would have been different. As it happens, this is indeed a good discriminant.

For further reading, I recommend the Wikipedia article on 'Clearing the neighbourhood', from which I took the data for the image. If you don't mind skipping over some technical bits, go for the original paper where this was proposed,

What is a planet? S Soter, The Astronomical Journal 132 no.6 (2006), p. 2513. arXiv:astro-ph/0608359.

which is in general very readable.