To endorse is to give support to someone or something. "I endorse this!" means "I think this is a good thing, and so should you."
People endorse in many ways. When someone endorses a politician, it means "You should vote for this person, and I'm putting my reputation on the line to say so." When someone endorses a product in a commercial, it means "Go buy this! You'll like it." To endorse is to give support. You can also endorse a check, which means writing your name or someone else's on the back so that you or the other person can cash or deposit it.