SaaS stands for Software as a Service. That means Software for which the business model is Subscription-based with recurring fees. The opposite would be on-Premise software which you buy once for an upfront fee but don't necessarily need to pay any recurring fee to keep using.
SaaS software are most often distributed through the internet and can be free as well. You could consider Gmail a SaaS software, Salesforce.com, Mint.com, NetSuite, Box.net also are all SaaS.
Microsoft Exchange for example, or Oracle and SAP are more often On-Premise software even if they push extensively to go SaaS right now as the business model is often times more attractive to small businesses.
If your startup is going to deliver it's service through the Internet, charging users a monthly fee (or offering it for free), it would certainly fall under the SaaS category.