1. Without referencing sheet music or a recording. Refers to the ability to play a piece of music. Wow, George is such a talented piano player! The way he can play pieces by ear after hearing them just once is so impressive!
2. By extension, in an improvised or flexible way. Typically used in the phrase "play it by ear." A: "Are we meeting at noon on Saturday?" B: "Around then. I have to run some errands in the morning, so let's play it by ear."
See also: ear