Earth is the only planet in our solar system that has copious amounts of water on it. Where did this water come from and why is there so much water on Earth compared to every other planet in the solar system?
The water was already present when the Earth assembled itself out of the accretionary disk. Continued outgassing of volcanoes transferred the water into the atmosphere which was saturated with water. And rain transferred the water onto the surface.
Compared to other planets and smaller solar system object Earth has a big advantage. It is large enough to prevent water molecules to leave the gravitational field, and it has a magnetic field which prevents atmospheric erosion (Wikipedia). This is due to the Earth's outer core being liquid (Moving charged liquid = magnetic field). Mars probably had oceans until its outer core solidified so much, that the convection was stopped. After the magnetic field disappeared a few million years of solar radiation removed all of the atmosphere and the oceans.
Your assumption that there is not a lot of water elsewhere in the solar system is incorrect. According the this article on NASA's website;
Missions in recent years have overturned our view of a dry solar system, returning mounting evidence of ample water from a vast array of locations.
Comets from the remote corners of our solar system are made of water and other ices. Orbiters, landers, and rovers reveal Mars as a watery world in the distant past -- a world that today may contain entire underground oceans of frozen water.
It also points out that
Jupiter's moon Europa has a frozen crust of water, covering a thick global ocean. By current estimates, it has twice as much water as all of Earth's oceans and rivers!
The perception that the rest of the solar system does not have a lot of water is probably due to the fact that the Earth is in the 'Goldilocks zone' where water can exist in all three phases at the surface of the planet. The temperatures on the rest of the planets in the solar system are either so hot that it can only exist in a vapor phase or so cold that it can only exist in ice at the surface or liquid beneath.
An important issue to realize is that water must have been very abundant in the protosolar disc, as tobias already stated. To expand on that I just want briefly touch upon the atomic abundances that we measure in the corona of the sun, as presented by a wikimedia-commons graph:
(Sidenote: Those abundances compare well to the revised Asplund2009-abundances)
We think that those numbers are representative of the bulk composition of the Sun, as it is only burning Helium since 4.567 Gyrs. Thus the composition is generally assumed to be primordial, or what the solar system started out with.
Now let us imagine this atomic mixture accumulating around the young terrestrial planets and focus on the most abundant elements H, He, C, N and O. In a thick planetary envelope, shielded by UV, equilibrium chemistry will set in. Then a lot of $H_2$ will form, He will stay inert and C, N, O will try to react with Hydrogen, simply because the number of encounters is much higher with that, than inside of the CNO group. Some CO will form, as this is a very stable molecule. But as C becomes depleted, and our protosolar nebula has $C/O \approx 0.5$, there is still a lot of Oxygen left. Thus it will inevitably combine to $H_2O$.
The result is, that we expect there to be really a lot of water around in Planet-forming discs.
The escape or destruction of water after this period however, is apparently also very efficient, and other answers have touched upon the retention of water on Earth. So in fact Astronomers at the moment rather wonder "where did all the water go?"
Water actually was never present on Earth before its creation and that is for sure.
Astronomers realized that there are two ready-made sources: comets and asteroids, the solar system’s gravel strewn among planetary boulders. The primary difference between the two is that comets typically have a greater concentration of ingredients that vaporize when heated, accounting for their iconic gaseous tails. Both comets and asteroids can contain ice. And if, by colliding with Earth, they added the amount of material some scientists suspect, such bodies could easily have delivered oceans’ worth of water. Accordingly, each has been fingered as a suspect in the mystery.
Adjudicating between the two is a challenge, and over the years scientific judgment has swung from one to the other. Nevertheless, recent observations of their chemical makeups are tipping the scale toward asteroids. Researchers reported last year, for example, that the ratios of different forms of hydrogen in asteroids appear to better match what we find here on Earth. But the analyses are based on limited samples, meaning there’s a good chance we’ve not yet heard the final word