You've really done it this time - you've upset her, and it needs to be fixed before further damage is done. Maybe you were late getting home, or forgot your anniversary. Maybe she's your co-worker, and you said something about the quality of her work that made her feel inadequate. She could be your sister, and you've really out your foot in your mouth about the way she raises her children. It could just be that you said something inappropriate to the woman at the grocery store checkout. Whoever the woman may be, and no matter your gender, you need to apologize, and make everything right in her world again.
Women tend to be based more on emotions than men, who generally see things with lessened emotion and more logic. Other women, especially, can be cruel to their peers and target their feelings in just the right way to hurt. However you've managed to hurt the woman you'll be apologizing to, it needs to be sincere and come from your heart. You'll need to first analyze what you said or did to make her upset; look within yourself and answer the question, "Why did I say/do that in the first place?" Maybe you weren't thinking, or were hurt yourself at something. Find the reason for your mistake before approaching the woman.
Sometimes an apology can be as simple as a sheepish smile and an "I'm sorry, do you forgive me?" Other times, actions might prove to be louder than words. Flowers and small gift items, especially if it's something related to her hobbies, work, etc., can do wonders in showing the woman you've hurt just how much you really do care. If you've upset a waitress, apologize with a bigger tip. Should you upset your mother, flowers or a card can really do the trick. And at all costs, try to apologize to the woman in person. A casual email or text message is not the way to go, as emotion isn't conveyed properly through these mediums. Even a phone call can make all the difference, if you can't be there face-to-face.
Always mean what you say in an apology to a woman. Most women are adept at detecting lies and insincerity in people, and you don't want to anger her more by pretending to be sorry when you're really not. An apology means so much more when it comes from the heart, rather than necessity. Never cheapen her feelings by saying she deserved the comment or action that hurt her, and recognize that she really was hurt, no matter how small you feel the hurt should be.
Apologizing to a woman can be tricky, but these few tips should help you to be in her good graces once more.