Basically, I have a story that would be best told through 'me' (the first-person narrator), telling the story to another character in the story, whom the story revolves around. Basically, the narrator would be named 'I', and the main character, whom 'I' is telling the story to would be called 'You' (not referring to the reader in a second-person sense). It would sound like: I longed for you, yet couldn't tell you. At least the style would sound like that. Hopefully the story itself has a little more substance. Does this style have a name? Will this style allow a reader to become involved in the story? Is this style any good? haha
It will force a barrier between the reader and the story for sure, but I think that's a good idea here. It will stop the reader from only allying themselves with the first person narrator and in doing so force them to consider his testimony more carefully and, therefore, interrogate the text more carefully. That said, it does depend a lot on your ability as a writer....certainly it will be more challenging than a typical (read: tedious) third person account, but the result could be promising if handled well...
Basically, it would read like a letter, then. That doesn't sound unusual or unorthodox, to me, but I expect it will be more challenging to write well. With skillful execution, it certainly appeals to the inner voyeur.
This isn't all that unusual, and comes up a good many times in fiction. Watson from Sherlock Holmes, and Nick from The Great Gatsby are two different examples. Watson is a secondary first person and has a close relationship with Holmes, even to the point that he may be unreliable and may be covering Holmes' faults. Nick, is a tertiary character to the Protaginist of Gatsby, recollecting his knowledge of the events, but was not very close with the man. A good number of Batman Stories are told from Alfred's recollection, as if the man is keeping a journal or writing a biography about Bruce Wayne, and the Superman/Batman series is both titular characters narrating the other's adventures (i.e. Scenes focusing on Superman will be narrated by Batman and vice verca) and works quite well to highlight their unusual friendship and diametric tactics in the ealier stories. The movie "Man of the Year", that one where Robin Williams becomes president, is told entirely through a secondary character's interview with Time Magizine reguarding the events leading up to Willaims' character recieving the titular award. So, it does have a name (first person doesn't mean the character is the protaginist, but rather, the narrator is directly involved in the action of the story and has limited knowledge of the events), it is used often, and it's just as good as any other style... that is to say, if the story works, it's good (I personally think Gatsby and Man of the Year were stinkers for different reasons, and I'm not a terriblely big fan of Batman.)
That would be an epistolary narrative form, except making the recipient of the "letters" the central character makes it uncomfortably like second person narrative. Epistolary narrative is somewhat out of favor these days. However, an example of that form would be Mary Shelley's Frankenstein. But the letters there are written to a party who is not part of the story.
Now there's a word I haven't heard before. As ever, you are a font of knowledge, Dave. Or is it just your mad google skillz?
It can be made to work. The only example that I've personally seen is in Jhumpa Lahiri's book Unaccustomed Earth, but she uses it well, and I enjoyed the story she wrote with it. (Look it up on Google Books. Better yet, here's a direct link. Start at the beginning of the second half of the book, entitled Once in a Lifetime.) There is no universal rule that says only some styles of writing work. Instead, there is a spectrum of what is easiest to work with and what "typical" readers most expect. You can work with anything in that spectrum: it's your right as an author. Just make sure you know how to handle the style you want to work with. You owe that much to your readers. Oh, and one last thing: Don't listen to anyone who tells you they can give you a "yea or neigh" answer to the question you posed after reading some short sample of your first draft. No one here is an oracle, and even if someone with some credentials tells you that there is another style that may (did you get the emphasis there?) be "better" or easier to work with, that doesn't mean it's right for your story, and certainly doesn't mean that you can't make the style you have envisioned work. Don't accept or reject advice ad hominem. Reason through it and make up your own mind.