How can communication with/control of satellites be maintained?

Our planet, Earth, is big, really big, and it just isn't possible to see all the way around it. So to keep in communication with satellites above it a number of different approaches are used depending on the type of orbit the satellite uses.

Satellites in a geostationary orbit maintain a fixed position over a certain place on the earth, and so can be directly controlled from anywhere that can see them.

Satellites that orbit the earth travel around the entire planet, and so to stay in communication with them a number of ground stations are used, and as the satellite passes out of site from one ground station it is picked up by another. There is usually some overlap. For example, a satellite might start over a base station on the west coast of the USA, hand over to a base station in Great Britain, and then hand over again to a base station is Australia, before directly communicating again with the base station on the west coast of the USA.

Another method can be used when a chain of satellites all in a similar orbit and operated by the same company need to communicate, there which ever satellite that can see a base station repeats the instructions to the other satellites in its chain. This way they all stay in communication via whoever has line of sight to the base station.