Nicolaus Copernicus (b. 1473, d. 1543), for primarily philosophical reasons, suggested a change in the Ptolemaic system: he placed the Sun at the center of the Solar system (a heliocentric system). Copernicus still subscribed to the notion of circular motions, so he too required epicycles. The Copernical system was no simpler than the Ptolemaic system, but it provided some regularity: Copernicus showed that the distances between planets increased as the distance from the Sun increased, and that the relative speed of planets in their obbits decreased going outwards.
A more detailed background is given by the Galileo project at Rice University.
Tycho Brahe (b. 1546, d. 1601) was perhaps the most accomplished astronomical observer ever. He built an observatory on the island of Hveen, north of Copenhagen, with a quadrant with a 31 foot radius (the quadrant you made has about a 2-3 inch radius). With this, Tycho made the most precise positional measurements made to that time of the planets. Tycho did not accept the Copernican heliocentric view of the solar system, but saw the advantages of its mathematical simplicity. He devised a geocentric system wherein all the planets (except Earth) orbited the Sun, and the Sun orbited the fixed Earth. Tycho is important not for this model, but for the quality of his observations.
Johannes Kepler (b. 1571, d. 1630) undertook the first real astrometric study of the planets. He was a mathematician, and did not make his own observations. Rather, he used Tycho's observations. He started by observing the motions of Mars. Because of the quality of Tycho's data, Kepler was able to separate the motion of Mars from that of the Earth. He then deduced 3 empirical laws that described the motions of the planets.
1. Planets travel elliptical paths, with the Sun at one focus of the ellipse. An ellipse (the oval in the figure at left) is a closed curve whose points lie a fixed total distance from two foci (the plusses). A circle (upper left figure) is a special case of an ellipse, where the two foci are the same point. This is a major change from all previous models of the solar system, which used circular motions and epicycles.
2. Planets move faster when closer to the Sun, and slower when further away. Specifically, the a line connecting the Sun and the planet sweeps out equal areas in equal times.
2. The square of the orbital period of a planet, in years, is equal to the cube of its mean distance from the Sun (in astronomical units). Mathematically, this is P2=d3.
In terms of describing the locations of the planets, and providing an empirical model of the solar system, Kepler's model is no better than the Ptolemaic model, but it is much simpler. There is a principle often employed in science called Occam's Razor, which states that given two equally good explanations of the same observation, the simpler one is generally preferred. Kepler's model clearly wins on that regard.
Isaac Newton made use of Kepler's laws when he formulated his laws of motion and universal gravitation. One can derive Kepler's laws from Newton's laws.