Link to the wiki page with all the info: Mystic Forge
The mystic forge is this big blue swirling thing in lion's arch. You toss in 4 items that go together, and get something else back. Generally speaking, you can throw in 4 items of the same quality and type (I.E., 4 "green" sigils, or 4 "blue" weapons/armor.) It has some specific recipes that come in handy too though.
You can toss in certain items, and always get a certain mystic weapon out. Here's the general idea:
For example, let's say you want a mystic rifle. You will throw in:
and your outcome will be: Mystic Rifle
This works for any other weapon, too. If you want a Mystic Shortbow, you put in 5 Ancient Shortbow Staff and 5 Hardened String instead of the barrels and stocks.
Something to consider before making one of these weapons: Check to see how much the pieces cost. I know in many cases, the weapon pieces cost alot more than they can be made for. Guild crafters will probably craft you the pieces if you mail them materials, so check to see if the materials are drastically cheaper. Sometimes you can save 5-10s per piece just by getting it crafted.