Band-Aid's name comes from the square band-aid she keeps over her cutie mark, which she has stated is "obscenely embarrassing". It is unknown exactly when she started calling herself Band-Aid, but her original name as a filly was "Glimmer Dusk". She usually wears human-like clothing and a collar.
Band-Aid has regularly expressed suicidal thoughts and a desire to disappear, but is kept from actually attempting the act by her obsession with Twilight Sparkle, who she met as a filly at her magical examinations. Twilight's awakening into her magic "drew [Glimmer Dark] to her", causing her to become infatuated with the other unicorn. Her sudden distraction caused her to lose control of her "dimming" and become visible, meaning Princess Celestia saw her sitting where she wasn't supposed to be. It is implied that her seeming exile from Ponyville might be connected to this, but it is not yet certain.
Her unique magical talent is the ability to make herself invisible, which she calls "dimming" herself. She has so far used this ability mainly to observe Twilight Sparkle from a distance and avoid detection by other ponies, but it was also stated that she sometimes uses her powers to steal food for herself from places such as Sweet Apple Acres. She states she only takes what she needs.
Band-Aid lives in the second basement level of the old factory between Ponyville and Froggy Bottom Bog, which once belonged to Stickyhooves Glue Company.