The Ptolemaic system used epicycles to make the orbits all come out to what they should around the Earth.

Coopernicus used only circles, but made the Earth go around the Sun, and then all of the planets go around the Sun as well. His system was more complex than the Ptolemaic. (And his justifications were extremely mystical points about what he thought was natural.)

The idea that orbits were elliptical is due to Kepler (who supported the Coopernican system). Kepler's 3 laws say:

  1. Planets move in orbits that are ellipses.
  2. The planets move such that the line between the Sun and the Planet sweeps out the same area in the same area in the same time no matter where in the orbit.
  3. The square of the period of the orbit of a planet is proportional to the mean distance from the Sun cubed.

(Note, you thought that the "weird elliptical orbits" were Ptolemaic. They were not.)

These laws were found to be very close, but not quite perfect. However Isaac Newton demonstrated that the second law implies that all forces on the planets are directed towards the Sun. The third law implies that the forces in question vary as distance squared. He verified that these two facts explain why the orbits are elliptical (actually they are conic sections). He found that if the force pulling us to the Earth also varies as distance squared, he could explain the Moon's orbit. He then showed that if he assumed that all pairs of masses attracted each other in this matter he could not only explain the above orbital facts, but he could explain a plethora of other phenomena, including tides, the precession of the Earth and the orbits of moons around other planets.

Cheers,
Ben