It's your lucky day. I'm an IT specialist who has woken up with bad period pains and using Mn to distract me!!
Guessing you want to know what it means as opposed to how to use a particular package?
Software as a service (Saas) It's basically 'pay as you go' for software applications. Also known as software 'in the cloud'.
In the old days you used to buy software and run it on a server or other computer. This meant that you had to pay up front for the software package (known as buying a licence) install it, configure it, and sometimes buy the kit for it to,run on as well. Every time the software got updated to a new version you would have to 'upgrade' , usually paying more money for a later version. It could be painful and expensive, and sometimes meant having to buy newer high performance kit for it to run on. It involved a lot of capital expenditure (capex) - money up front which IT departments and to plan, estimate and budget for. Then fight the CFO to get money for! Examples include SAP.
In Saas the software comes with the 'full stack' ie it runs 'in the cloud' on kit /servers owned by the Saas company. So this means that you don't need to buy and own big heavy duty kit to run it on. All you need is an Internet connection to access it. Another benefit is that you don't ever need to pay to upgrade the version you use as the SaaS people automatically give you the latest version (mostly anyway). In most Saas (as with most cloud) you pay for it as you use it and can switch it off at any time. Prices tend to be much lower per head than trad software. It's become very popular for these reasons, and also because it is classed as 'operating expenditure(opex) in accounting terms, which is beneficial to the way company accounts work (there is no asset purchase or ownership so no depreciation applies). Disadvantages include inflexibility ( you don't own it so can't always do as much configuration to shape it to your exact needs). Example Salesforce.com
Other terms you may come accross related to this are
Iaas - infrastructure as a service (eg aws the Amazon cloud) where you pay as you go to use someone else's processing power and/or storage
Paas - platform as a service . Again borrowing someone else's kit but they also manage it for you.
Hope this helps. If not feel free to ask more as I'll be online periodically tomorrow.