Real orbits actually aren't quite elliptical. Pure elliptical orbits arise when
1. There are just two objects around.
2. The attractive force between them follows the inverse square law, like Newton's gravity.
3. There are no other forces around.
4. The objects are perfectly spherical.
5. Newton's laws are valid in a Euclidean space.
All of these conditions are pretty nearly true for planetary motion.
1. Although there are several planets, they usually aren't close enough to each other to exert strong forces.
2. and 5 are excellent approximations for our solar system, though measurably not quite right.
3. Is a very good approximation, since electromagnetic forces are relatively unimportant for planets.
4. Is a fairly good approximation, although the Sun's non-spherical shape (caused by its rotation) does affect the orbits of inner planets.
Anyway, given the simple inverse-square force in a simple space, Newton was able to derive the elliptical shape of the orbits by pure math. The derivation is not especially simple- I always forget it and have to struggle to recreate it. There's a nice version in one of Feynman's books.
(published on 10/22/2007)