A process is basically a program in execution. It consists of a program, its data, its ownership information (UID), the current position at which the computer is in the programs execution and all the other information needed to run the program. The UNIX operating system will occasionally decide to halt a process running and run another process. This gives every process its fair share of processing time. A process is also halted if it is waiting for some I/O operation to finish, like accessing a printer.
When a process is halted the operating system has to store all the information about the processes current state, so that the process can later be restarted exactly where it left off.
Processes can start other processes. These child processes have exactly the same ownership and permissions as the parent processes. Parents can control their children and send them signals. The operating system can also send signals to processes. Their are a wide variety of signals, and most programs react to signals by dying. There are a number of programs though, which are mostly operating system related and take more intelligent action towards signals sent to them.