Imagine you are in a party in England and someone approaches you and says: "How do you do?", what is the correct answer to that? As crazy as it may sound, the correct answer would be: "How do you do?" This expression is very formal but it is still used in the UK.
On the other hand, if you were in a party in the United States and you were asked the same question, you could easily answer "Fine, thank you" or "Not bad!". Americans don't understand the use of "How do you do?" the English have.
"How are you doing?", although it is mostly American, can be used in any informal situation. But, be careful, don't say that to your teacher or your boss!!
In case you don't know what to say, a simple "How are you?" or "Nice to meet you" is always alright. Don't get nervous!
