Generally the term "tracking" refers to determining the object's location. Often this is done via ground based radar "tracking stations" maintained by various organisations like the US Air Force. Typically the process after launch is that the launch vehicle provider sends us (the satellite operators) and the Air Force an initial estimate of the spacecraft's orbital elements, which contain the position and velocity information. The Air Force will then publish an updated set of orbital elements as soon as their radars or other tracking assets (they have some telescopes and other devices too) have determined a solution.
As the Air Force makes more observations, the solution improves.
We satellite operators use these approximate solutions to point our radio antennas on the ground (ground stations) toward the place the satellite is supposed to be. Sometimes we will find the spacecraft right away, but sometimes it's not exactly where we expect so we then have to search for it. Often this means we point the dish a little above the horizon where the satellite is expected to rise or set and we just wait for the satellite to fly through our radio beam.
Information such as the timing and strength of the received radio signal are then fed back into models to improve the position estimate of the satellite and this information is shared with the Air Force to ensure they are tracking the correct object. This is most important if several satellites are released in the same launch.
Many satellites in Low Earth Orbit (LEO) also contain special GPS receivers that can precisely locate the spacecraft and when available, this data is sent down via radio to the ground station and generally provides the most accurate tracking data, which again is shared back to the Air Force.