What makes an antagonist for a story? Quite simply, the force that opposes the protagonist. The antagonist could portray that of a person, an animal, a storm, a situation and so on that serves as the conflicting force for the protagonist. But, a villain is something far different. A villain is the antagonist, but an antagonist is not always the villain. Did you get that? This is similar to a hero vs. a protagonist. The protagonist is the main character, not necessarily the hero. Just ask Edgar Allan Poe or William Shakespeare for that matter. A villain doesn't just oppose the main character, but goes so far to cross that line and become a serious danger to all that can be considered "good." There is always a villain in a "good vs. evil" archetype story. The villain is the bad guy. The hero is the good guy, right? Well, it depends on perspective. One hero is another man's villain.
To make a good villain for a story, you need to find conflict. Why is the villain bad? What does he/she/it want and why is it a bad thing? Like every hero needs a backstory, so does every villain. These villains serve as the polarizing force to the hero, so they must be dynamic and show some sort of self-righteous that almost mirror's the hero in some way. A villain should be able to defend their actions and almost convince the reader that they belong--that they are worthy of their title. In many ways, a villain should outshine the hero. If the hero is any hero at all, they'd be able to reclaim their audience by displaying their true spirit to do what is right, regardless of their own desires and doubts. The villain does not have this self-control. Villains are selfish, calculating, hateful, etc. To be more clear, they are the urges of humanity that are unchained, unrestricted. Heroes are selfless, or they are supposed to be. If you would like to really differentiate them: The hero will make the self-sacrifice. The villain could give a damn.
What's the best part of it all? The best part is when an antagonist becomes the villain. The final fall from antagonist to major bad guy is a shocking and enjoyable experience for readers. This is when the opposing force acquires bad ideas that become dangerous to all of us. Usually the villain makes the hero, but the hero cannot make the villain in the same way. Basically, if your story has a protagonist (but not necessarily a hero), please don't include a villain (to oppose the hero). It wont really gel. An antagonist could be many things, but a villain is a figure. Here's the difference:
Hurricane Darth Vader
Abusive parent Ultron
Bully at school Voldemort
Corrupt boss Joker
Opposing team The Devil