Why are the orbits of planets in the Solar System nearly circular? Except for Mercury, the planets in the Solar System have very small eccentricities.
Is this property special to the Solar System?  Wikipedia states:

Most exoplanets with orbital periods of 20 days or less have near-circular orbits of very low eccentricity. That is believed to be due to tidal circularization, an effect in which the gravitational interaction between two bodies gradually reduces their orbital eccentricity. By contrast, most known exoplanets with longer orbital periods have quite eccentric orbits. (As of July 2010, 55% of such exoplanets have eccentricities greater than 0.2 while 17% have eccentricities greater than 0.5.1) This is not an observational selection effect, since a planet can be detected about equally well regardless of the eccentricity of its orbit. The prevalence of elliptical orbits is a major puzzle, since current theories of planetary formation strongly suggest planets should form with circular (that is, non-eccentric) orbits.

What is special about the Solar System that orbits of planets here are nearly circular, but elsewhere they are moderately or highly eccentric?
 A: Just wanted to supplement the answers already posted with a few notes re: exoplanet eccentricity. In my understanding, the reason why exoplanets have a median eccentricity ~0.3 vs. almost circular orbits in the solar system is not quite satisfactorily explained just yet 
(this paper is still my favorite simulation that attempts to address the origin of eccentricities).
1) The relative importance of disk-planet and planet-planet interactions is not understood completely.  It is not clear whether Type I/Type II migration have an important role in shaping the final configuration at all. This is indicated by the number of planetary systems with large inclination wrt the star's equatorial plane discovered recently. 
2) There is probably some degree of bias in e-Np histograms, given that there is a degeneracy between a single-planet eccentric solutions and multiple-planets circular solutions to radial velocity observations. See, e.g., this paper and others:

finding that (1) around 35% of the published eccentric one-planet solutions are > statistically indistinguishable from planetary systems in 2:1 orbital resonance, (2) another 40% cannot be statistically distinguished from a circular orbital solution, and (3) planets with masses comparable to Earth could be hidden in known orbital solutions of eccentric super-Earths and Neptune mass planets.

3) Planetesimals and resonance crossing have played an important role at some point in the history of the solar system re: the eccentricity evolution of Jupiter & Saturn (a recent paper about this). Why this led to low eccentricities in the solar system might reside in the relative configuration and mass ratio of J & S, the specific mass in planetesimals in the disk compared to other protoplanetary disks, etc. 
4) Large eccentricities, especially with small planets, tend to lead to instability and scattering, so invoking the anthropic principle to some degree is not totally unjustified...
A: This property is likely shared by every other planetary system belonging to the same class as ours. Anthropically speaking, orbits with higher ellipticities will have more extreme environments and thus will be less likely to harbor life. As for the physical reason, I'd guess you could come up with a thermodynamic argument. Intuitively, an elliptical orbit seems to be further from equilibrium than a circular one. A collection of bodies orbiting a star, with high ellipticities initially will then relax to a state where the orbits are less and less elliptical. Of course, this is all heuristic.

Edit: In response to @Marek's and @MSalters comments let me add some clarification.
When a planet with some eccentricity $e$ interacts with a transient object (asteriod etc.) which process is thermodynamically more likely: that the planet eccentricity increases or that it decreases as a result of the interaction? The LRL vector is a conserved quantity but not when the object is in a noisy environment. The interactions between the planets themselves are also greater when they have orbits with high $e$. As they slowly lose $e$ by scattering and damping events, the mutual interaction will decrease as the orbits become more circular.
A: So far, this is an unknown question that is the subject of current research. For example:  
Sean N. Raymond, David P. O'Brien, Alessandro Morbidelli, Nathan A. Kaib, "Building the Terrestrial Planets: Constrained Accretion in the Inner Solar System"  

To date, no accretion model has
  succeeded in reproducing all observed
  constraints in the inner Solar System.
  These constraints include 1) the
  orbits, in particular the small
  eccentricities, and 2) the masses of
  the terrestrial planets -- Mars'
  relatively small mass in particular
  has not been adequately reproduced in
  previous simulations; 3) the formation
  timescales of Earth and Mars, as
  interpreted from Hf/W isotopes; 4) the
  bulk structure of the asteroid belt,
  in particular the lack of an imprint
  of planetary embryo-sized objects; and
  5) Earth's relatively large water
  content, assuming that it was
  delivered in the form of water-rich
  primitive asteroidal material.

http://arxiv.org/abs/0905.3750
A: The short answer is tides.
Maybe there're more shaping effects for the sun, but at least that's the primary effect shaping moon's orbit to be circular. 
The thing is that the tidal torque on the sattelite has a strong dependence on the distance. Secondly, the orbital mechanics motion has an interesting property: if you give a bump to an orbitting body it will still pass thru the same point (but it won't get as far out). This is, qualitatively, how tides slowly shape off outer parts of the orbit. 
However, the effect would be the same for other types of friction strongly depending on the distance, for example friction between the bodies' athmospheres.
A: This was previously a comment to space_cadet's answer but became long (down-vote wasn't me though).
I don't understand space_cadet's talk about unstable orbits. Recall that two-body system with Coulomb interaction has an additional $SO(3)$ symmetry and has a conserved Laplace-Runge-Lenz vector which preserves the eccentricity. Because interactions between planets themselves are pretty negligible one needs to look for explanation elsewhere. Namely, in the initial conditions of the Solar system.
One can imagine slowly rotating big ball of dust. This would collapse to the Sun in the center a disk (because of preservation of angular momentum) with circular orbits and proto-planets would form, collecting the dust on their orbits. Initially those planets were quite close and there were interesting scattering processes happening. The last part of the puzzle is mystery though. If there were still large amount of dust present in the Solar system it would damp the orbits to the point of becoming more circular than they are today. The most popular explanation seems to be that the damping of the eccentricity was mediated by smaller bodies (like asteroids). Read more in "Final Stages of Planet Formation" - Peter Goldreich, Yoram Lithwick, Re'em Sari.
A: A short answer is that dissipation (e.g. dust, gas interactions with planetessimals) is good at removing energy from a system, but not angular momentum. Circular orbits have the minimum energy for a given angular momentum.
For short-period exoplanets, the primary form of dissipation is tidal forces of the star on the planet (similarly, the moon is on a nearly circular orbit around the earth, although it certainly didn't start that way!)
