How do I make my 12 year old child apply lotion on his dry skin?

Rewards work. The preteen/teen years are the two's recycled. If you want to avoid win/lose confrontations don't set yourself up for them. Make allowance contingent on you seeing him/her apply the lotion, or figure out another reward. If you want the behavior you must understand the kid's developmental stage and work around it.
By now... ...I'm sure you realize that you can't make a 12-year-old do much of anything he doesn't want to do. Nobody likes to feel powerless, including children; refusing to do what you ask gives him a feeling of power, and unless the dry skin bothers him enough he'll probably continue to refuse. If you can persuade him to do it, make sure he applies the lotion after bathing, while his skin is still moist.