Why does the Earth not have rings like Saturn or Uranus?
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
Earth does have rings! Not icy ones like Saturn, but ones made up mostly of solid masses a few meters across. A relatively sharp ring is centred on geostationary orbit. A rather more diffuse one (although probably with higher total mass) reaches up to altitudes of a few hundred km from the surface. The second one is constantly losing mass back to the planet due to friction from the atmosphere, but is replenished by a complex biological process.