From the time you were a little kid, you were probably taught to say "I'm sorry" if you upset or hurt a friend. When you're young, that's often enough to fix things. But as you grow older, an apology isn't always enough. Occasionally, it doesn't work at all. Be genuine, sincere and acknowledge if your actions hurt your friend. Hopefully, she'll be willing to accept your apology, but be prepared if she doesn't.
Keeping It Real
If you're apologizing just because you think you should, but you don't really mean it, your friend might very well sense this. An insincere apology is almost as bad as no apology at all and it might add insult to injury. If you've already hurt or upset your friend, don't make it worse by offering an apology you don't mean. Be sincere when you apologize, own what you did that caused the rift, and let your friend know that you're aware your actions hurt her. Don't spend a lot of time explaining or rationalizing your behavior, and don't make excuses for yourself.
Timing Is Everything
Even if your apology is entirely sincere, your friend might not yet be ready to accept it. The hurt may be too fresh or her anger too raw. Seeing you again might be an unwelcome reminder of what happened, and she might not be ready to move on, even if you are. She might still be processing what happened, or she might want you to have to suffer a little longer to ensure you realize the extent of her pain or anger. Offer a sincere apology, but acknowledge her feelings if she can't accept it right away. Tell her you understand and give her some time and space if she needs it.
Dealing With Betrayal
Your apology is likely enough to mend a rift caused by something minor. If, however, your actions resulted in your friend feeling betrayed, what you consider a small rift might seem like an uncrossable chasm to her. If she believes you betrayed her confidence or "stole" her boyfriend, for example, all the apologies in the world likely won't make it right in her eyes. Such betrayals, whether actual or perceived, can leave deep and lasting wounds, especially if the two of you were close friends. Earning back her trust is something that will happen only over time, if at all.
Your friend may choose not to accept your apology because she simply doesn't want to. Maybe the two of you have a history of miscommunication or problems and she's decided it just isn't worth trying to fix the friendship. Perhaps she's moved on to other friends and no longer wants a relationship with you. Or maybe the friendship was never that strong in the first place, and the recent spat just gave her the excuse to stop being friends. Sadly, this likely leaves you few options but to move on yourself.