An endorsement is a form of public support or approval. Endorsements are given to politicians and products.
If you give something an endorsement, you're basically saying "I approve of this person or product." Celebrities give politicians an endorsement if they think you should vote for them. When celebrities do commercials for products, those are also endorsements. If someone puts their fame or name behind something, they're endorsing it and giving their approval. That goes for checks, too: when you sign one you've just written, you've endorsed it.