If you want to accept the invitation you could say, "Thank you very much, yes I would like to go to your party" or "Yes, thank you very much. I would love to go to the movies with you." If you don't want to accept you could say, "Thank you but I am sorry I can't make it that day." or "No, I can't go because I am very busy." If it is someone you would like to go with on another day you could say, " No, I can't make it on Monday but can we go another day?" or even more advanced: I can't make it, a rain check for another day?

I see some good answers already but I'd like to address the "formal" part of your request. Some formal responses might be: "Thank you for the invitation and I would be delighted to attend your birthday celebrations" or "Thank you for the invitation but I must regretfully decline because I will be on a business trip that day".

Invitations? Like to a party? Or a date? In person or in writing?