If you want the dictionary definition, just double click on any word.
"How do you do?"
"How are you?"

If I had a Euro for every time someone got this one wrong - I'd be a rich bunny!

How do you do?

This is not a question. It is another, very formal way of saying "Hello." It is also very British.

The correct response is; "Pleased to meet you." or "How do you do." or just "Hello."

We only really use it the first time we meet someone.

How are you?

This is a question, but the person asking it doesn't really want to know the truth about your aching back or hangover.

A polite response is; "I'm fine thanks. And you?"

For more on this topic see here.