1.1  NEUTRAL INTRODUCTIONS

Remember, if you learn the neutral form correctly, it should be acceptable in almost any situation . except perhaps for introducing royalty, which should not come up very often!
Think of yourself as addressing only one person at each moment. Picture yourself looking and talking first to one and then the other. First, you are dedicated to giving information about John Brown to Mary Smith. After that, you become dedicated to giving information about Mary Smith to John Brown. Do not try to multiplex the information!
Okay, let' s try it.

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1.1.1  ADDRESSING THE FIRST PERSON
What you want to say to Mary Smith can be divided into three parts:

1. Mary Smith's name2.  Your purpose for coming to her3. John Brown's name

So what you say to Mary Smith could be:

______________Mary Smith,  I'd like you to meet someone.   This is John Brown. ___________________________________ Mary Smith,  I'd like you to meet a friend of mine.   This is John Brown.  ___________________________________ Mary Smith,  I'd like to introduce a colleague of mine.   This is John Brown.  ___________________________________ Mary Smith,  may I introduce someone to you?   This is John Brown.

Normally, you would never address Mary Smith with both of her names. You do so in this situation so that John Brown can hear them. 

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1.1.2  ADDRESSING THE SECOND PERSON
What you want to say to John Brown can also be divided into three parts:

1. John Brown's name2. Mary Smith's name 3. Information about Mary Smith

So what you say to John Brown could be:

___________________________________John Brown, this is Mary Smith.  She is a former colleague of mine.  ___________________________________ John Brown,  this is Mary Smith.   She works for our competitor.  ___________________________________ John Brown,  this is Mary Smith.   She is a good friend of mine.  ___________________________________ John Brown,  this is Mary Smith.   We went to school together.

At this point it is time for them to greet each other, so whatever information you give about Mary should be very short! After they have spoken their greetings, you can add information about either one of them to help the conversation get going.  

1.1.3  PUTTING BOTH TOGETHER

_____________________________________________________________ Mary Smith,  I'd like you to meet someone.   This is John Brown.      John Brown,  this is Mary Smith.   She is a former colleague of mine. _____________________________________________________________ Mary Smith,  I'd like you to meet someone.   This is John Brown.      John Brown,  this is Mary Smith.   She works for our competitor. _____________________________________________________________ Mary Smith,  I'd like to introduce a colleague of mine.  This is John Brown.      John Brown,  this is Mary Smith.   She is a good friend of mine. _____________________________________________________________ Mary Smith, may I introduce someone?  This is John Brown.      John Brown,  this is Mary Smith.   We went to school together.

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1.2  INFORMAL INTRODUCTIONS
Informal introductions are more common and much simpler. We will give you a pattern to follow, but you should know that it is almost impossible to make a mistake with informal introductions because there really are no rules! For example, it is no problem to introduce a woman to a man or an older person to a younger person. The reason is that EQUALITY is such a very strong value in American culture the same as informality is.
With informal introductions, it is okay to follow the same pattern as with neutral introductions. However, because it is an informal situation, you should address each person by his or her first name.

_____________________________________________________________ Mary,  I'd like you to meet someone.   This is John Brown.    John,  this is Mary Smith.   She is a former colleague of mine.

With a slightly more informal situation, you can leave out the last names altogether.

_____________________________________________________________ Mary,  I'd like you to meet someone.   This is John.    John,  this is Mary.   She is a former colleague of mine.

And in some extremely informal situations (for example, parties), it is okay to give just the first names!

_____________________________________________________________Mary,  this is John. John, this is Mary. _____________________________________________________________ Mary,  John.    John,  Mary. _____________________________________________________________Mary, John.

It cannot get much more informal than that!