Eyesdropping is essentially eavesdropping with your eyes. When someone looks at something that is not theirs. The thing that the person is looking at does not necessarily have to be confidential but it just has to not be theirs like a computer screen or a cell phone.
"Raymond, stop eyesdropping, look at your own computer screen."
"Liz, don't you have better things to do than to eyesdrop?"
The act of looking at another person's Instant Messenger or text messaging conversation. Much like eavesdropping only slightly more annoying. Generally done by parents, younger siblings, nosy friends, etc.
Ahhhh! God! Stop eyedropping on my convo! Stupid sister...