You might be surprised to learn that yes, terrestrial planets can have rings like Saturn. As you probably know, most objects in the solar system are held together by their own gravity, and every object is a satellite to some more massive object. The force between two objects depends on the distance, and when the distance between the objects is much larger than the size of the objects, then the force is approximately constant across the entire satellite. However, if a satellite orbits too closely to its host body, the force is much stronger on one side of the satellite than the other. If these forces are very large, they can literally rip the satellite apart into thousands of pieces, which rearrange themselves to form a ring around the object.
So how close is too close? It depends on the size of the host body and the density of both the satellite and the host body, but it is normally around twice the radius of the host body (called the Roche limit). If a moon were to orbit inside of this range, it would break up and form rings, regardless of the type of planet. This leads us to your second question: since the Earth is way smaller than Saturn, the region around Earth where an object can fall in to break up into rings is much smaller than the region for Saturn.
Now you might be (correctly!) thinking that there is more to the story here. After all, the Earth was hit by a meteorite 65 million years ago that killed the dinosaurs; that didn't break up into rings. Also, there are videos
of the comet Shoemaker-Levy 9 smashing into Jupiter in 1994, which also doesn't form rings. There are two factors at play here. First, for a planet to induce a breakup, the satellite must be held together by its own gravity. That is the case for celestial objects large enough to become roughly spherical, but not for small comets and meteors. If an object is not spherical, this is an indication that it is held together by their own chemical bonds, which are stronger than the gravitational force for a very small object. Secondly, if an object does break up, that doesn't mean the object will necessarily form rings around a planet. When the object breaks up, the net momentum of the object will be unchanged. If the object was moving extremely quickly during its breakup with respect to the planet, most of the pieces have enough energy to overcome the planet's gravitational force and will escape into the solar system. Only when the pieces are moving at just the right velocity will they form rings around the planet instead of crashing into the planet or escaping.
All four gas planets in our solar system have rings, although Saturn's are the most dramatic. It's unclear to scientists why the rings are able to stay in a stable orbit (without either escaping or crashing into the planet) over long timescales. It is partly believed to be caused by "shepherd moons" which lie in orbit near the edges of the rings. These moons protect the ring material from leaving the system: gravitational forces push the material into the ring when it tries to escape, or accretes the material onto the moon itself.
Extrasolar gas planets are likely to have some rings for the same reason that solar gas planets do: their Roche limit is pretty far away from the surface, giving a large region for satellites to fall into and break up, forming rings.
As a final note, if you are curious as to why the Roche limit for an object, Wikipedia's Roche limit page
has a detailed derivation, starting with Newton's Law of Universal Gravitation.
Thanks for the great question!
(published on 01/10/2011)