I put them into two major classifications: communicators and sensors.
Within communicators there are comm relay satellites, and signaling sats. The former receive a signal from the ground and relay it to another location in some way. Some examples are ground station to ground station, like geosynchronous comm birds, ground station to distributed receiver, like direct broadcast satellite, or store-and-forward LEO like Orbcomm. Signaling satellites include navigation satellites like GPS or GLONASS.
Remote sensors have some sort of device for detecting some signal and relaying it to the ground. Some examples are: GOES and TIROS weather satellites, which detect reflected light or other emitted radiation from the atmosphere. Visible spysats, which look at the ground Chandra X-ray Observatory, which images deep space in the X-ray bands THEMIS, which measures the magnetic field that it is in. Signals intelligence sats, which listen for interesting radio. (I differentiate these from com sats in that the person generating the signal isn't trying to get it relayed to another listener.)
Another orthogonal classification system could be made based on the type of orbit the satellite is in. (Orthogonal meaning that any of the above could be in any of the following orbits.)
LEO = low earth orbit, generally less than 12 hours GEO = geostationary earth orbit, a 24 hour equatorial orbit HEO = high earth orbit, more than 12 hours, but not GEO Solar orbit = no longer in earth orbit.
These could be further subclassified, but that's probably enough.
Different satellites are at different heights. They can be anything from hundreds to thousands of miles above us.