Fellowship Draft was probably my favorite play format in this game. The rules are basically as follows:
1. Everyone has to bring their own Ring-bearer, One Ring and site path to the table before the draft even starts. This is to ensure that you'll actually have the cards needed to play the game. And, yes, it means that you have to pick your sites without knowing what your deck will be like. For this reason, you might end up drafting sites during the draft, once you start to figure out what your deck is going to look like.
2. Everyone gets a draft pack. You open your pack and get a look at what kind of cards you'll have for a basis for your deck. These cards are NOT passed or drafted--they're yours to keep, and they'll usually give you some sort of starting theme for your deck. (i.e., you'll get a bunch of Isengard Orcs, or Moria archers, or whatever.)
3. After everyone has reviewed the contents of their draft packs, they all open a booster pack. In the example of Fellowship draft, you'd each open a Fellowship pack. Everyone selects one card to keep and passes the rest of the pack to their left. You continue in this fashion until the entire pack is gone.
4. Everyone opens their second pack. (In Fellowship draft, it would be a Mines of Moria booster.) Do the same process as step 2, except you're passing to the RIGHT this time.
5. Repeat one more time with a third and final pack. (This example = Realms.)
6. Once all cards have been drafted, you get some time to build your decks. The deckbuilding rules are a little different than in standard format, though--the minimum deck size is 40 instead of 60.
7. Now you're ready to play! The game plays just like a constructed game, except that once your draw deck is gone, you have the option of reshuffling your discard pile into your deck. You can do this only ONCE per game.
EDIT: I forgot to mention: the booster packs you use are determined by the block of your draft pack, of course. But you don't HAVE to do one pack from each set of that block; for example, in a Fellowship draft, you COULD do 2 Fellowship/1 Mines, or 3 Fellowship, or the classic 1 Fellowship/1 Mines/1 Realms, or whatever. Just make sure the block of your boosters matches the block of your draft packs!