An application is a program which you interact with on the desktop.
A process is an instance of a particular executable (.exe program file) running.
A service is a process which runs in the background and does not interact with the desktop. In Windows, services almost always run as an instance of the svchost.exe process, the windows service host process; however there are sometimes exceptions to this.
Sometimes, processes may run in the background without interacting with the desktop, but without being installed as a service.
Antivirus programs usually employ a service so they can continue running even when the user is not logged in.
Processes usually exit when an application is closed, however this is not always the case. Some programs, particularly download and backup programs, may continue to run in the background without displaying any windows. Antivirus is also an example of this - in addition to using a service, many antivirus applications run a process silently in the background which only displays an application to the user when action is required.