I'm so sorry that a scheduled exam kept me from your party. What bad luck! I would have so much preferred to celebrate with you.

The common word when you can't do something because the schedules overlap is to refer to a schedule "conflict." How to phrase the apology depends on how formal or informal your relation with your friend is. A short thing to say, e.g. in a text message, is to say "I'm sorry, I can't come because I have a conflict with an exam." Whether to say more than that is up to you. The important thing is to let your friend know so that they will have an accurate "head count"--they know how many people will be coming, and how much food to get.

You can apologize to your friend for not being able to attend her birthday party by being honest and considerate in your communication. Here's a suggested message you can send: "Hey [Friend's Name], I hope you're doing well! I wanted to talk to you about your upcoming birthday party. I'm really sorry to say that I won't be able to make it this year. My final exam is scheduled from 7 am to 9 am, which means I won't be able to join your celebration from 8 to 10. I feel terrible about missing your special day, and I wish I could be there to celebrate with you. Please know that it's not because I don't want to be with you on your birthday; it's just an unfortunate scheduling conflict that I couldn't avoid. I promise to make it up to you and celebrate your birthday in a different way once my exam is over. I hope you have an amazing birthday party with all our friends, and I'll be thinking of you even though I can't be there in person. Let's plan something fun for when I'm free after my exams. Once again, I'm really sorry, and I hope you understand. Enjoy your day, and let's catch up soon! Best wishes, [Your Name]" This message conveys your apologies, explains the situation, and expresses your desire to make it up to your friend in the future. It also shows that you care about her and value your friendship.