To calumniate is to make a false accusation against someone or spread lies about how awful they are. Don’t calumniate your rival in the race for class president, because when the truth comes out, you’ll be the bad guy.
The courts would say to calumniate is to slander. Everyone else would just say it’s mean. Calumniate is just a fancier way of saying “lie” — but specifically lying to tarnish someone's reputation. If your sister says you purposely broke a vase and you didn't, she calumniates you. If you get in trouble in class, you could calumniate to the principal about your teacher being a wretched mess — but the other students might speak up and prove you wrong.