Honorific term for someone's eye(s). Used when speaking respectfully about another person's eyes, often in set phrases like お目にかかる (to meet) or お目が高い (having a discerning eye).
See also: 目
Your eyes, Sensei, are very beautiful.
You have a discerning eye. This is indeed a fine article.
Could you please look over this document?
His work caught the editor's eye.
audience (with a superior); interview · first public appearance; debut; coming out · trial service (of a servant); probation · audience with a shogun (Edo period); right to such an audience
scolding; telling-off; dressing-down
eyes
chaperoning; monitoring behavior · watchdog; chaperone; watcher
Honorific prefix お (御) + 目 (め, 'eye'). The prefix お adds politeness and respect.