SaaS is the applications layer of Cloud Computing delivering software capabilities to organisations of all types through the users web browser.
The SaaS Vendors host a single, centralised, multi-tenanted version of the applications  software. Each customer accesses their own instance and the SaaS vendor ensures the users data is held in a secure environment. The users pay a fee  for the SaaS software either by periodic subscription, per transaction charge (Pay as you Grow) or based on usage time.
SaaS applications have been developed for email, team collaboration,HR and Payroll and accounting and the SaaS model is now being used to provide high performance, highly resilient and secure applications across a range of company sizes and industries.
Here is an excellent video describing the advantages of deploying SaaS applications by salesforce.com.
They key advantages of SaaS include:
Eliminates need for on premise computer hardware and traditional up-front software licence fees
No Capex, software supplied on a “pay-as-you-go” basis as an Operational Expense item
Less reliance on IT department for support as this becomes the responsibility of the SaaS vendor
Increased operational agility as business users can quickly subscribe to new services through their web browser – in most cases they can test the SaaS applications in a 30 day trial period
The software is delivered as a utility and usage can be scaled up or down
Ease of use, the SaaS software is designed to be intuitive with online help and support services
Flexible access for office-based or mobile users via any internet connected PC or Smart Phone
Closer supplier relationship to ensure customer loyalty. The SaaS subscription model makes it easier for customers to switch vendors.
The SaaS vendors continously improve their applications based on their knowledge of how the customers really use the applications. The software updates are applied frequently (out of hours) and can be accessed by all the users when they next log in.