According to Emily Post, the overall rule is that one person is always introduced to another. This is achieved either by the actual use of the word to -- "Mr. Johnson, I'd like to introduce you to Mr. Smith"-- or by saying the name of the person to whom the other is being introduced first, without using the preposition to. An example of this is: "Mrs. Gartner, may I introduce you Mr. Wagner."
In addition to the overall rule, there are three basic rules:
1. A man is always introduced to a woman. Example: "Mrs. Stein, I'd like you to meet Mr. Brown." or "Mr. De Marchi, may I introduce you to my mother, Mrs. Wamser."
2. A young person is always introduced to an older person.
3. A less important person is always introduced to a more important person. This rule can be complicated, since it may be difficult to determine who is more important.
Following Mrs. Post reccommendations, do not introduce people by their first names only. Always include a person's full name. Avoid calling only one person "my friend" in an introduction. It implies that the other person isn't your friend.
When you introduce yourself, don't begin by saying: "What's your name?" Start by giving your own names: "Hello, I'm Anna Elisabeth..."