Dander is an informal term for a material shed from the body of various animals, similar to dandruff. It is composed of skin cells. It is a cause of allergies in humans. Older etymology has it that dander is a rare (perhaps Yorkshire) synonym of dandruff (OED). In the idiom of "getting dander up" the colloquial meaning, anger, is probably unrelated to the above usage.
Pet dander is generally worse in older animals than in younger animals, as older animals slough off more skin cells than younger animals. Dander is microscopic, and can thus be transported through the air. Through the air, dander can enter the mucous membranes in the lungs, causing allergies to some individuals.