How many natural satellites does the Earth have?

True satellites, only one – the Moon.

Although… actually the Earth-Moon system is complicated, it is an intermediate case between a pure binary system (like Pluto-Charon) and a pure primary-satellite system (like Mars and its moons, or Jupiter and its moons).

And in fact, both in terms of mass, size and distance from the primary, the Earth-Moon system is closer to being a binary system than a conventional “primary/satellite”… but well, let’s leave it at that.

The Earth also has some very interesting co-orbitals.

  • 3753 Wheat it appears to be orbiting the L4 point of the Sun-Earth system in a strange kidney-shaped orbit. It is actually a co-orbital of our planet, but the complex and interesting interaction it has with the gravity of the Sun and the Earth put it in this 1:1 resonance. Unstable in the long term, but at least several hundred thousand years.
  • 2010 TK7 This asteroid is much more interesting still. Yes, it is actually orbiting the L4 point, but the plane in which it orbits this point is perpendicular to the plane in which it moves around the sun, so it moves through the cosmos in a kind of cosmic corkscrew
  • 2002 AA29 co-orbital with the earth in a “horseshoe orbit” (so called because of the shape similar to the letter “C” that they take when viewed from the solar north pole co-rotating with the Earth), their orbit is in plane different from the plane of the Earth’s orbit, so it goes through the cosmos in a permanent corkscrew
  • (419624) 2010 SO16 another co-orbital in a horseshoe orbit – and a particularly stable one according to the simulations, will be there between 120 thousand and several million years
  • (277810) 2006 FV35 remarkable because although its orbital period around the sun is almost exactly one year, its highly eccentric orbit crosses the orbits of Venus, Earth, and Mars

Other asteroids are temporarily trapped by Earth’s gravity and orbit it for a while, only to later exit the gravitational well (and return to orbit the Sun) or crash into our planet.
An example is 2006 RH120, which orbited the Earth from Sep/2006 to Jun/2007 before “breaking free”

The Wikipedia article on Claimed moons of Earth and Cruihne’s (linked above) talk about many of these asteroids in orbits with the same period as Earth’s.