Completely Web Based
This help desk software is installed on your web server, and only a web browser is needed to use it.
Local Software with Web
This help desk software requires local installation on each server or workstation, but also has some web integration.
Remotely hosted is a help desk service provided by another company. Usually this is also completely web based, but the difference being these companies provide everything for you for a monthly fee or per ticket basis. The software resides on their server.
This help desk software requires local installation on each server or workstation.