"Eyes" are the spies who work for the government and are situated throughout Gilead. Anyone could be an Eye, and the assumption is that characters are always being watched. It's really interesting that the visual aspects of spying are emphasized when you consider that the Handmaids, in particular, are supposed to be kept from both seeing and being seen. As reproductive objects, they must not be sexualized, and one of the freedoms Gilead is supposed to provide them is freedom from the lascivious male gaze. But the watching the government does, through the Eyes, is even more invasive.