As long as I've known my best friend, she's been complaining about these tiny red bumps on the backs of her arms. She thought they were some kind of acne, like what you get on your back when you workout, so she went to her dermatologist to find out.
It turns out that it was a common condition called keratosis pilaris - a complicated name meaning bumpy skin. Actually 50% of people have it, so if you don't have it yourself, you probably know someone who does. It can be on the backs of your arms, your tush, and your thighs. The dead skin cells that are supposed to just flake off, get plugged up in the hair follicles. They look like tiny red or white pimples that have a dry, sandpaper feel. They don't hurt or itch, but they can make you feel self-conscious about baring your skin.
What can you do about these bumps? Use a good moisturizer twice a day - one containing salicylic acid, urea, or alphahydroxy might be your best bet. You can also take hot showers, let the skin soak a little, and then gently exfoliate using loofah gloves. A dermatologist can also prescribe a topical retinoid cream (made from vitamin A), which may help promote cell turnover and prevent the hair follicles from getting plugged in the first place. Whatever you do, I'm sorry to say that it takes constant maintenance - if you stop, the bumps will come back. Since they're harmless, I wouldn't worry about them too much.