To catch somebody's eye is to get them to look at you. You lift your chin very slightly at the bar, and the barman nods back.
The possessive changes to fit whoever it is — catch the waiter's eye, catch her eye, catch my eye. It runs two ways round, and the difference is who is trying: a person catches your eye on purpose to be served, and a dress in a window catches it without meaning anything at all. Eye-catching is the adjective made out of the second kind.
1phrasecatch somebody's eyeTo attract somebody's notice.
“You have to catch somebody's eye before they will serve you.”
catch somebody's eye is an act of drawing notice.
Practised in Common Word Pairs