An elastic band worn around the upper arm. Originally, this was to allow for adjusting a sleeves length to better fit the wearer, or to hold up very long gloves that have a loose fit around the arm.
On Danbooru, arm garters are usually worn like an armband, and are not used to hold up a sleeve.
Sleeve garters were used by men during the 19th century when men's ready made shirts came only with extra long sleeves, thus requiring the garter to adjust the sleeve to the appropriate length for the wearer. This kind of garter is still commonly seen in some of the outfits worn by barbershop quartets.
For women's gloves, a garter-like object was used to hold up long gloves around the bicep of the arm. These were used for long gloves that had a loose fit around the arm, such as some of the formal wear gloves used during the Napolenoic period.