A pen name is a fictionalized name that some writers use instead of their real one. Sometimes writers do this to keep their different kinds of writing separate, using, for example, a pen name to publish children's books, and the real name to publish adult horror books. It's a perfectly legitimate practice, as long as you're not using it for an unlawful purpose.
You can usually include this information in a query for books or a cover letter for journal submissions. Some writers will sign the letter with their pen name and indicate in the body of the letter that they've done this. Or, you could simply include your pen name after the letters a.k.a. (also known as) under your legal name on your letterhead or wherever you include your contact information. Other writers will use their legal name to sign letters and mention their desire to use a pen name in the body of the letter.
Which option you choose is up to you, but it's important to be clear and up front with publishers. You don't want to give them reason to question your intentions. And they will need your legal name in order to pay you.