Someone who is round-eyed has eyes that have gone wide and round — usually for a moment, because something has shocked them, frightened them, or left them staring in wonder. It is a look, not a face shape. A child can be round-eyed at fireworks; a shopper can go round-eyed at a huge price. Writers like the word because you can see the feeling on the face without naming the feeling. Wide-eyed is very close; round-eyed leans a little more on that open, almost circular look, as if the eyes themselves had become circles. You will meet it more in stories than in everyday talk. Though actually, you would not call a person round-eyed as a permanent description of their eyes (that would just be “big eyes” or “round eyes”). This word is about the moment the eyes open wide. It is ordinary English, not slang and not stiff, but it does sound a little bookish if you drop it into a chat.