What do we mean when we say it today?
Have you stayed up late and gotten loads of things done? Do you like working at night when others are asleep? We hate to break it to you, Shmoopers, but you're a night owl.
Nowadays, a night owl is anyone who stays up late. Usually people use it when referring to someone whose doing something at night other than sleeping (like the rest of the world.)
Of course this has very little to do with what Shakespeare meant by the phrase. But it just goes to show—sometimes things take on a life of their own.