Do other planets in our solar system experience eclipses or is this unique to Earth?
Pretty much every planet with a moon can have eclipses - you'll have seen photographs of Saturn with shadows cast from its moons, like this one from wodumedia.com:
If you were in a balloon where that shadow is, you would see a solar eclipse. It wouldn't be as exciting as from Earth, as the Sun would appear so much smaller from this distance.
If you want the planet to eclipse the moons, this happens far more often than on Earth, as the moons are so much smaller than Saturn. They often are completely shadowed.