By desktop, it simply means on a Mac or PC someone downloaded the app from iTunes (which stores in your iTunes until you connect your device and sync). Sometimes when you are at your computer it's easier to grab something on sale from iTunes even if you don't sync. The app is still flagged as bought, and if the price goes up you don't have to pay anything.
When browsing or grabbing lots of apps, the computer is usually a bit faster to navigate and easier to flip to app reviews and videos.
For us developers it's an irritating category because we don't know if they are getting the app for an iPad, iPhone or iPod touch.