Not so long ago, in that same rather near world, there was a tower in Deadwind Pass known by the name of Karazhan. It was the first step in what promised to be an epic raiding scheme, more accessible than ever with the new 10 and 25 man raids, versus the previously very constricting 40-man ones. Home to Tirisfal's Last Guardian, Medivh, it was packed with all sorts of lore and touted as a dungeon even from WoW's release (but didn't make it in 'til BC). However, it (like almost all BC's raids on release) required the completing of a quest chain, to obtain the key which opened the gate before Karahzan. Later the attunement feature was removed - only one raid member needs to have the key, to open the door which lets you enter Karazhan.
Who can do it?
There are a few requirements before you can get the key to enter Kara. First, you will need to have access to a Key to the Arcatraz, which comes from the quest chain that begins with 'A Heap of Ethereals'.
Next, you will need access to Shadow Labyrinths, which requires a key from a chest in the end of Sethekk Halls. (Both of these doors can also be avoided by using a rogue with 350 or greater lockpicking).
Lastly, you will need access to the Black Morass, which requires continuing the quest 'To The Master's Lair' through 'Escape from Durnholde'.
To actually do the chain, you will need to be level 70. However to do Karazhan solo, you must be level 80 and fairly well geared - as well as knowing what you're doing. Even then, It'd be difficult.
Where to begin...
You can either start with the lead-in quest, Archmage Alturus, which begins with Apprentice Tasserel in the World's End Tavern in Shattrath. If you don't want to go there for some weird reason, you can always just head out in front of Kara and pick up the two beginning quests.
Who can tell you how to do it?
There's a great guide on Wowwiki that takes you through it step by step. (A note on Wowwiki: It's a wiki, so, things can be wrong on the site! That, however, does not make it less creditable - as long as you note the discrepancy, and fix it yourself.)
First, you get a great deal of xp and gold, as with any long quest chain. However, all through this quest, you get to take your first steps through the Violet Eye reputation, which has a great deal of neat rewards if you keep at the raid for a while. Really only useful if you plan on staying at lvl 70, though, because most if not all rewards will be replaced within the first few zones in Northrend.
Next, you'll get your very own copy of 'The Master's Key', which will allow you to enter the tower of Karazhan at any time you like.
Lastly, if you continue the quest all the way through the dungeon, you get a 'Violet Badge', a decent trinket with a ton of arcane resistance - useful for soloing a number of the bosses in Kara, especially The Curator. Who is indeed equipped for gallery protection.
Why in the gorram world does this quest matter?
Well, I'll tell you. Firstly, as with all others, this could be removed in Cataclysm - we just don't know - and it'll be neat to have a unique, no longer obtainable key if that is so.
Secondly, this was the first rite of passage for all BC raiders. It's a cool thing to be able to know the roots of raiding, and what has come before - so you can feel lucky that there are no more fights with ridiculously annoying RNG like Prince, or feel sad about the loss of your free epix off of Chess. Also, having a key is nice in case you want to shoot for the Karazhan achievement - with nobody doing it nowadays, it'll get harder and harder to find a key.
Lastly, it's just a pretty cool quest. All sorts of lore is intertwined throughout.