It's well known that Kepler worked out his laws by fitting curves to Tycho Brahe's data on the trajectories of planets through the sky. What was this data? How does one record the trajectory of a planet through the sky? A pair of angles measured from some reference point? What sort of reference point - a steeple on the horizon? Also, how did Tycho Brahe keep accurate enough time to know he was recording the position of the planet at equal intervals of time?
Today, sky coordinates are measured as "Right Ascension" (RA) and declination. These are similar to the angular coordinates we use for the Earth's surface but are measured on the celestial sphere relative to the celestial equator and pole. By using the current sidereal time, it is possible to map the local sky coordinates (ie. a horizontal bearing relative to true north, and an inclination above the horizon) to RA & Declination.
Brahe used devices such as Quadrants to measure this inclination. There's a engraving of one mounted to the side of a building (presumably at Uraniborg) at http://en.wikipedia.org/wiki/Uraniborg (Brahe did most of his observations at Stjerneborg due to problems with the Uraniborg instrument mountings).
As for time, the clocks of the day were accurate enough because the escarpment had been invented a couple of centuries earlier. A sidereal clock can be reset every night (using a known star and a meridian) so it only needs to be accurate for 24 hours. Temperature changes will be relatively limited in this period (and minimized further by resetting the clock twice a night), and there will be no motion. Yes the clocks available to Brahe would be have been inaccurate compared to Harrison's chronometers of a few centuries later but Harrison was designing clocks to last months without adjustment and to survive violent motion and significant temperature changes.
Meridian: A great circle line that goes through the location (eg. Uraniborg Observatory) and the poles. It would be a line due south from Uraniborg. A star crosses the meridian when it appears to be due south of the celestial pole.
Sidereal Clock: A sidereal clock is one that is synchronized with the stars and not the Sun. The Earth actually rotates in one sidereal day (about 4mins short of a normal solar day) but because it has moved a little bit in its orbit, the usual point of reference (the Sun) has moved slightly, hence the solar day is slightly longer than a sidereal day.
The modern solution would be to use a solar clock and then apply a correction based on the calendar (trivial with a microprocessor). But if you're dealing with mechanical clocks, it is just as easy to set the pendulum to run slightly fast.
At the time of Brahe and Kepler they did not use the right ascention and declination to record the movement of planets. These coordinates are related to the Earth, and it is known since the times of Hipparchus and Ptolemy that one has to use the ecliptic coordinates, that is a system related to the Sun motion. (Ecliptic is the large circle in the sky on which the Sun moves. All planets move in the planes that are close to the ecliptic plane).The corresponding coordinates are called (celestial) longitude and latitude.
Brahe's data for the planets consisted (roughly speaking) of positions of the planets (their longitudes and latitudes) at some known moments of time. Because the latitudes of all planets are small, one can consider the motion in longitude independently of the motion in latitude, which simplifies the problem. This motion is not uniform (and even not always in the same direction), so the problem was to create a kinematic mechanism which would "explain" this motion. Such a mechanism is known from the antiquity, but it gives a small disagreement with the observed data. It is in his attempts to explain this disagreement (in the case of Mars) that Kepler made his great discoveries (the first and second laws). Much later he also discovered the 3-d law. The most fundamental discovery was that the orbits are not combinations of uniform motions on circles, but a motion on an ellipse according to the "law of equal areas".
This remarkable discovery was probably the greatest revolution in the history of astronomy.
Here one can find a bit more explanation: