Computer communications can be simple - a cable between two computers. Or it can be very complex - one computer connecting to another over the Internet on the other side of the world. Many different media can be involved - phone lines, data lines, microwaves, radio, infrared, or satellite uplinks.
Computers are digital but phone lines are analog. A modem converts the signal between these two types of transmission. Digital modems don't actually have to translate, but are called modems anyway - cable modems, ISDN, and DSL.
Computer needs software for communicating. A dialing program connects to an analog modem. FTP program transfers files. Terminal emulation convinces a network that your computer is one of its own terminals. Encryption keeps your data safe from prying eyes and ears as it is transmitted.
A network lets multiple computers share hardware and files and even programs. A LAN is a local area network. A WAN is a wide area network.
Many networks are in a star configuration where one computer is the server and handles all of the communication between the computers. It may handle most of the processing itself. A bus network connects computers and other hardware to a communication line. Each computer can contact any other computer or device that is connected to the network bus line. A ring network connects all computers in a circle. When one computer is down, the network is down. A router or bridge connects two networks together.
A file server receives requests for files and sends the whole file (no matter how large) to the computer that asked for it. That computer does the processing to answer the question and prepare a report.
A client server receives a question and uses it own processor and files to answer the question and send the answer back to the computer that asked.