It's a brace for the elbow, and is used to support the elbow if you're having joint problems. You can also use it to support your wrist and ankles if you're having wrist or ankle joint problems. (Basically they reduce stress on your joints, especially when exercising or doing manual labor.)

(If they donated blood, they'd have a bandage, similar to this.)

They also make similar braces for your knees, though those often are in black. (Knees take more force, so you need a different design. But it's all the same principle.)

Source: I have both wrist and knee problems.