- The unusual 3753 Cruithne (our "2nd moon" and pronounced "krooy-nyuh"). See here or for a very neat animation here.
- An old Apollo 3rd stage was briefly thought to be our "3rd moon", here.
- Even more unusual object is 2002 AA29 which is our once and future "moon", here.
- And a quasi-satellite called 2003 YN107, here.
Some of these orbits are very weird! Crutithne is not actually pushed away from Earth. The annimations are usually in two dimensions and really cannot capture the complex interactions that happen in the four dimensions of space and time.
There may also be other asteroids in stable solar orbits near earth at the five "Lagrangian Points". These are strange places in space where you can get gravity without actually having a large mass like a planet. These happen when you have two objects with a large (planet size) and really large (sun sized) mass interact. Rather than trying to think about the math, you could think of a standing wave in a river as an analogy.
Update: There have been some historical beleif in and attempts to find moons around Earth. A summary can be found as part of this longish essay, here.