The part of your clothing that covers your arm (or part of your arm) is a sleeve. Some t-shirts have long sleeves, others have short sleeves, and your tank top doesn't have any sleeves at all.
On a hot day, you might roll up the sleeves of your shirt, or you might change into one with short sleeves (or no sleeves at all, also known as sleeveless). Another kind of sleeve is a container or protective covering, like a record sleeve or a sleeve of crackers. If you have something "up your sleeve," it means you're hiding something to use later in a surprising way.