SaaS is the hosting of software applications and accessed using the internet via your web browser.
It is a shared (multi-tenant) service to many users in much the same way that electricity is sold. The computer software is usually developed by and the infrastructure managed by the service provider (which is Saasu in this case).

SaaS offerings can be accessed from any PC, Mac or Linux machine connected to the internet using a web browser. In effect you are accessing the software we developed on our servers (by using your web browser) instead of installing software on your PC. This allows us to buy much more expensive and powerful technology and recover our investment over time in very small payments from many users. You get a lot more value for your subscription without the upfront investment (because we have already made that investment on your behalf).

There is also a comprehensive explanation of SaaS on Wikipedia.

Compelling arguments for choosing a “SaaS” utility over old style “On Your PC Software”

• Secure - your data is managed in highly secure data centres versus old style accounting software which tends to be on a laptop, desktop or office server.
• Ease of deployment - Software installation isn’t required.
• Lower cost deployment - Usually no capital outlay.
• Access - Anywhere, anytime, multi-browser and multi-user capability. Mac, PC or Linux friendly.
• Flexible licensing - User pays model of a small monthly fee, no large up front costs.
• It’s fast, run reports in seconds
• Concurrent access to a single database globally
• Support costs are usually lower due to technical issues with software install and upgrade not needing to be covered by the software provider.
• No version clashing. Old vs newer versions of the same software.
• No sync or copy problems. Everyone is working on the same data, there is not risk of old or wrong versions of files being used and no need for copying them around.
• Licenses issued can be used on an computer, not locked to one PC.
• No network required for multi-user.
• Auto-backups daily.
• Auto-upgrades each release. Better application life-cycle management.
• Because an application is ‘live’ it can do a lot more automated tasks than software can. When your PC is off your software can’t do much for you.
• No need to email files home to work on them. Everything is available in your browser