Is there an order of importance in introducing people? For example, exactly how do I introduce my boss to a new co-worker?
Yes, there is an order of importance when making an introduction. An easy way to remember is to say the name the person with the most authority first. For example, if your boss's name is Jane Doe, and you are on a first name basis, then simply state, "Jane, this is Sam Smith, our new accountant; Sam, this is Jane Doe, our CEO."
If you wish to be more formal, you can state, "Ms. Doe, I'd like to introduce Sam Smith, our new accountant; Mr. Smith, this is Ms. Jane Doe, our CEO."