What is an Antagonist?
Before you can create a good antagonist, you have to understand what exactly an antagonist is. On a basic level, the antagonist is the bad guy. This is the person who gets in the way of the protagonist (the good guy). The antagonist is the reason (or one of them) the protagonist doesn't get what he or she wants. Sometimes the bad guy is a direct threat to the good guy, but sometimes not.
Your antagonist can take many different forms. Usually the antagonist is a separate and distinct character intent on doing the protagonist harm. Other times, he or she will simply get in the way, foiling the plans of the good guy (either intentionally or by accident). Occasionally, the antagonist is not its own character, but a part of the psyche of the protagonist. This is rare and not all that easy to do, so unless you've been practicing, don't do this.
Creating a Believable Antagonist
One of my biggest problems with most novel is the haphazard way in which the bad guy is treated. The good guy (and all his buddies) get to be fully-formed characters, but the bad guy exists as a two-dimensional chunk of mud. If you want me (or anyone) to read your work, you need to put just as much effort into creating your antagonist as you do any other character.
Start by understanding exactly what the role of the bad guy is. Most bad guys don't have to carry the weight of the story (unless your antagonist is also your main character). The protagonist lurks in the background, waiting to get in the way. This doesn't mean he (or she, but I'm going to stick with he) doesn't have to be as real as anyone else in your story, but it does make your job easier if you don't have to worry about him on every single page.
The key to creating a realistic antagonist is to give him a story. Why is he the way he is? In my first novel, Arianna's Tale, the bad guy (who happened to be a girl) was the way she was because she was horribly jealous of her brother. This jealously drove her to destroy everything he had. She wasn't born evil, but her father was so focused on her brother, and her brother was so focused on anything but her, that she slowly slid down a very slippery slope. And ended up dead because of it (but that was at the end of the novel, of course).
In my second novel, the bad guy had originally been a good guy. But he was hopelessly corrupted by an even worse guy. That antagonist also died. Antagonist three was a man whose biggest failing was his attitude toward women. His father had raised him to believe they were nothing, not worth the ground they walked on. This attitude eventually led to his abusing the wrong woman. Women, actually.
And in my fourth (and current) novel? The antagonist is a woman, an enchantress. Except that she'd not exactly real. Instead, she's a construct, a creation whose sole purpose is to carry out the bidding of her master (who we won't meet until book five). But she's innately curious, and this actually helps the protagonist get out of a rather nasty situation.
The point is this: your bad guy (or girl, as the case may be) must have a story. There must be a reason for your antagonist to act the way he does. Most antagonists weren't born bad. Most of mine certainly aren't. Maybe the exception is the construct, but only because she was created by a bad guy. So when you're creating an antagonist, make sure they have a history and make sure you at least allude to that history. A good antagonist (ignore the oxymoron) will be more than just a bad guy. He will have a unique history that has made him who he is.
