A process is any piece of software that is running on a computer. For example, your anti-virus software runs in the background as a process, which was automatically started when the computer booted. Some processes start when your computer boots, others are started manually when needed.
Some processes are services that publish methods to access them, so other programs can call them as needed. Printing services would be an example of a service type of process, where your email program can just call the print services process to say it wants to print, and the service does the actual work.