Why Earth does not have rings?
A moon is held together by its own gravity, and pulled apart by the tidal action of a planet. If a moon comes too close to a planet it will be ripped apart by the planet's gravity and become a ring. The closest a moon can come to a planet is known as the Roche limit, and it is dependent on the mass and density of the planet and moon.
A large planet, such as as Saturn, has a large Roche limit, and has collected many moons over its history, some of which have been ripped apart to make rings.
The Roche limit for the Earth-moon system is at a radius of about 10000 km, which is very close to the surface of the earth (about 6400km). And unlike the gas giants, terrestrial planets don't have many moons.
So with few moons, and small Roche limits, the opportunities for ring formation are rare. That said, Phobos, one of the satellites of Mars, is now so close to Mars, that it is held together as much by its rigidity as by gravity, and is slowly getting closer to Mars. Within 50 million years it will be ripped apart and form a (small) ring around Mars
Just thought I'd add that when you say "very close to the surface of the earth" we're still talking about a distance that is beyond Low Earth Orbit (LEO)--it's just very close to Earth's surface compared to the distance of the Moon.
1) Another interesting question here (that I should be able to find out with a little research) is why Jupiter doesn't have more massive rings than Saturn (large magnetic field?). 2) I propose that if mankind makes it through the current global warming/ over population/ etc. crisis that there may be rings around the earth in another millennia.
@JackR.Woods Rings form under specific circumstances. Saturn's outer E ring seems to be fed by enormous jets and volcanism from Enceladus but that doesn't explain the inner rings. A disintegrated moon is perhaps likely what formed the more densely concentrated inner rings. If Io drifts inside Jupiter's Roche limit, it could break apart and give Jupiter glorious rings, even bigger than Saturns, but I have no time-table on when or if that might happen. The 4 moon system in orbital resonance might be long-term stable.