Always remember: a cell phone is a radio.
A cellular telephone is basically a two-way walkie-talkie that acts like a telephone. With a walkie-talkie, you either talk or you listen; with a cell phone, you can talk and listen at the same time. You can dial a number to place a call. You can receive calls. You can do fancy things like three-way calls, conference calls, call hold, and voice mail.
All conversations on cell phones are unprotected and can be intercepted. Don't think of a cell phone as a telephone, think of it as a radio. Sure, there are laws to protect you against illegal eavesdroppers, but obviously these laws are hard to enforce since it's hard to catch someone listening in on your call. Be careful of what you say (or transmit with a modem) on a cell phone!
Cellular phone systems can be "analog" or "digital". Older systems are analog and newer systems are digital.
Each cell phone identifies itself to the cellular system each time it places or receives a call so that the cellular system can verify it is a valid paying customer. The cell phone's identity includes the phone number that is assigned by the service provider.
Why is it called "Cellular"?
Many different types of systems have provided city-wide two-way communications, such as radio systems for taxis or the police. In these systems, a single antenna is located near the center of the city. Each two-way conversation occupied one channel so if there were 100 channels in the city, only 100 simultaneous conversations could be held. To cover the entire city, the antenna is placed on a tall building and emits a very strong signal.