"The Eye" is an body expression that is usually used in a topic about girls. "The Eye" is when a girl's eye dilates ( The pupil gets bigger ). When this happens this means the girl sees something she likes.
Here is an example from a song by Common called "Punch, Drunk, Love." feat Kayne West.
" Hey baby am I crazy or was you giving me the eye? You said maybe, well you crazy because I know I'm too fly... "