On an electric bass guitar or indeed any electric guitar, the term "pickup height" refers to the distance between the pickups and the strings. Electric guitar pickups create a signal from the vibration of the strings interfering with the electromagnetic field of the pickup. Since this is the case, adjusting the distance between the pickup and the strings affects the sound and playability of the bass in many ways.
The adjustment of the pickup height is in itself a very simple operation. There are two screws on either side of the pickup. When the screws are turned one way the pickup rises from the body of the guitar (closer to the strings) and when the screws are turned the other way they are adjusted away from the strings.
The difficult part is knowing what height to adjust them to, and this depends on many factors all of them relating to personal preference.
When pickup height is very low (close to the string) the pickups will receive a stronger signal from the strings. This results in a brighter, louder tone from the guitar and allows the guitar to be played with a lighter touch. If you’re playing style includes a lot of finesse and softer tones, you will probably benefit from lowering the pickup height.
However, if you play a lot of slap bass or fast-driving heavy-metal bass you may find the strings frequently coming into contact with the pickups. This can result in "clatter" noise as the strings slap against the pickups. Some pickups also emit a loud pop when this happens, particularly if the wiring in the bass is not grounded properly. In this case you will need to increase pickup height to eliminate this problem.
The general rule of thumb is that you want the pickups to be as close to the strings as possible without actually touching them.
Another consideration is using pickup height to adjust the sound of the bass. Many basses naturally produce a louder tone from the low strings (E and A) than they do from the high ones (D and G). This can be compensated for by adjusting the pickup height unevenly giving more height under E and A than under D and G. Or you may prefer a muddier sound from your bass which can be achieved by lowering the pickups and increasing string height.
The best advice is to try it several different ways. Adjust them really high and try that for a few days, then try them really low and see how that goes. If you have multiple pickups do one high and one low and see how that plays for awhile. Only through experimentation can you find out what really works for you and your bass.