You use presents when you're referring to gifts, like all of those meticulously wrapped pairs of tube socks your mother leaves under the tree for you each Christmas. Presents can also be used as a present tense indicative third-person verb that means gives someone something or shows oneself.
You use presence when you're talking about the state of being present or the way that someone carries him- or herself, as in Of course Shanice got the lead in the school play! She has incredible stage presence.