I'm a non native speaker of English. In our learning we were told that when we say hello to someone we use equivalent phrase:
How do you do?
In response we do say:
How do you do?
But I'm unsure about the usage in the USA or UK. I haven't heard from them. I'm an Indian.
Is this the correct way of introducing someone?
You are absolutely correct. "How do you do" is an old fashioned introduction and is an obsolete synonym of "hello", and consequently the proper response is "how do you do?".
Mr Darlington: How do you do, Mrs Windermere?
Mrs Windermere: How do you do, Mr Darlington?
This exchange is exactly equivalent to the more modern (and at the time, more vulgar):
Mr Darlington: Hello, Mrs Windermere.
Mrs Windermere: Hello, Mr Darlington.
Note that the phrase "how do you do" has long since gone out of fashion in British English, and was never in fashion in American English.
Generally speaking, native speakers in both British English and American English now use the phrase hello instead, although how do you do does still persist in some formal settings in British English, generally during formal introductions, and lives on as howdy in some dialects of American English (and like how do you do, the correct response to howdy is howdy).
Don't be surprised if using this phrase that you receive odd looks - the phrase is excessively polite, and may easily be mistaken (even by native speakers) for the more common introduction "How are you?", to which the proper response is "I am fine, thank you, and you?".
How do you do? is a bit formal sounding; at least here in the U.S., we'd usually use the more informal How are you? instead.
How are you? (or, even more informally, How ya' doin'?) is indeed used as a greeting as often as it is as a genuine inquiry, although there are, of course, some exceptions. If I had been in an accident, for example, and was paid a visit as I was laying in my hospital bed, I would not answer "How are you doing?" with another "How are you doing?" Instead, I'd assume the person wanted to know how I was feeling after the accident. However, when I pass someone in the hallway at work, it's not uncommon for one "How're you doing?" to be answered with another, as if we had said "Hello" instead.
Of course, this generalization can vary somewhat based on region and local culture, a fact that was brilliantly parodied in this commericial, where the visitor doesn't seem to recognize that, at least in New York, "How ya' doin'?" is not usually an open-ended question (as it might be in Texas, where the visitor is apparently from).
Usually, when "How are you doing?" is given as an answer to "How are you doing?", that's done among friends and close acquaintances. However, when being introduced to someone for the first time, we generally answer the question, rather than repeat the question, as Bill said in his comment:
How are you doing?
Fine, thanks – and you?
Something like that would probably be the "safest" response if you weren't sure what to say.
In my region (Upstate NY, American English) it is a little different. We do answer the question, although the answer can vary.
The customary greeting is "How are you?"
The response:
What you never do is admit you are not good in a formal contact. Even if you are having a very bad day, when a stranger (other than your doctor) asks you how you are doing you use one of the first two forms. Even at funerals, the bereaved often say they are "doing ok", "managing" or "getting by" rather than admitting how they really feel.
The "how are you" question can also be used to expand a brief contact. If someone says, "good evening" you can respond "Good evening to you too. How are you doing this evening?" The first person then has an opportunity to further expand the conversation or shut it down with a simple formal reply.
It would be a perfectly normal introductory exchange in formal situations in the UK.
Possible snarky responses to the stuffy, outdated "How do you do?":
Indeed, how do I do what I do? I even amaze myself sometimes, not to mention others. I just have that magic touch.
How do I do? I make like a swimming duck: I appear calm and composed, but below the surface, I'm paddling like hell.
How do I do what? You mean how do I in particular do it, or how is it done?
Okay, seriously now. "How do you do" used to be a greeting. It asks the person about their well-being and the progress of their daily affairs. However, it is no longer grammatical for this purpose. In modern English, what person is doing is an ongoing activity, and that requires a progressive tense which is formed using a gerund. The correct grammar is "how are you doing?" The grammar "how do you do" does not refer to an activity which is going on now, and the second "do" requires an object. For example:
Help me! How do you do this math problem?
How do you do everything, with your demanding job, and all those other commitments?
"How do you do" not followed by anything is now strictly an idiom (or a "canned phrase", if you will) which is used only in formal introductions, and is not equivalent to "how are you". It means the same thing as "pleased to meet you". You can respond with "how do you do", or with "pleased to meet you" or "nice to meet you".
"How do you do" is often contracted to "how'd you do" and spoken quickly. It sounds very similar to the contraction of "how did you do".
Never use "how do you do" on someone you have already met before. If someone whom you have already met says it to you, refer to the snarky responses above.
Lastly, "do you do" could occur without an object in some contrived sentence. For example: "I heard you don't study for exams. How do you do?" This asks the person how he or she performs on the exams in spite of not preparing. Because the exams are not going on now, and it is just a general question about the person's study habit, it is not appropriate to use "how are you doing".
If someone says to you:
how do you do
you may simply reply with:
I am [fine / OK / good], and you?
and don't repeat the greeting verbatim.
