Check how much damage you caused
Before getting in touch with other people, try to remember what happened the night before. You should be able to access the damage before apologising to anybody. Cell phones are extremely handy when it comes to knowing the extent of your stupidity. Delete any inappropriate Tweets you may have made the night before. In addition, check your text messages along with the pictures in your phone. This will let you know who you should say sorry to for your bad behaviour while being drunk. Try to act quickly and say sorry directly to the people you might have hurt.
Face to face apology
Now that you have identified all of the nasty behaviour you did the night before in a drunken state, you should have a list of the people that you need to apologise to. Make sure that you personally meet each person and do a face to face apology. This will show that you are being sincere and that you actually mean what you say. Try to apologise quickly and do not start talking about each and everything that happened the night before. The point is to apologise to each person. Remember, never apologise over the phone or through email as this will look very impersonal on your part.
It might be a good idea to call all of the people that you want to apologise to over for lunch or some social gathering. Remember, keep it light and refrain from alcohol. Bringing all the people together will show them that you are sincere and that you can make light of the situation.
Ensure it does not happen again
The most important part when apologising after getting too drunk is to ensure that your bad behaviour does not happen again. Let everyone know that you will refrain from alcohol or at least limit your intake to help control your behaviour.