Microsoft On Demand is a new type of program utilization. Similar to Video On Demand and other popular use-as-needed products, this version of Office provides an ultra-lean option for productivity.
The idea is, rather than having full versions installed, you are using the program only when needed. But Office on Demand actually goes a step further. Rather than even opening up a full program as needed, you are only accessing those parts you are currently using.
Is anything left behind once you log out of the On Demand site? Not much, but cache files are left behind. Microsoft indicates these are necessary for performance considerations.
Here is a quick overview to get you started with Microsoft Office On Demand.