The Taken King is the new expansion for Destiny that raises the level cap and adds new content. As of now, The Taken King is ready to be played, assuming you have the latest patch downloaded. Check out this article to get more details.
Destiny: The Taken King Available To Play Now
I really hope most of you have already downloaded the 2.0 patch because if not you have a long wait ahead of you. There was a new patch right before launch that came in at about 100Mb. Do not be alarmed if you see 17gigs again, this is just to finish the patch off and open up The Taken King for players to access. I know, I freaked out a little bit too when I saw that number.
The Taken King adds a new sub class for each class, new equipment and new quests to the game. There is also new PvP modes and maps and a new raid coming this weekend. Overall the expansion wants to bring Destiny to the next level. You can find the new quest on Mars.
Check this site for more Destiny: The Taken King Guides!