These new quests will task you with making the newer Cataclysm items in order to change to a different Alchemy specialization. There have also been improvements in the switching process. Now when you want to unlearn your current specialization, you simply click the little red symbol next to the current specialization in your Professions window to remove it:
Then talk to an Alchemy trainer to select a new quest for the specialization you want. The big improvement here is that all three specializations rely on crafted goods. No longer will you be expected to run through old Burning Crusade dungeons for any of the Alchemy quests. Not that this is a huge burden for max level characters, but I think changing to crafted goods is more appropriate.
Each of the quests will require you to bring back a variety of items that are relevant to the specialization you selected. Here are the quests you can choose from:
You can see that the requirements vary pretty wildy on each of these quests in terms of the value of the goods to be traded. I believe MMO-Champion reported on this aspect of the change. I assume the intent is to curb the number of specialization switches that occurs among the player base. At this point I think most of you realize that you need to choose one specialization and stick with it. Although honestly, I suspect the vast majority will stick with Transmute Master.
One thing I noticed when trying out these quests is that it currently does not count crafting actions. You can use items you already have in your inventory for the quest. Personally I think a crafting quest should require you to make the items, which is how our Jewelcrafting daily quest currently functions. Another wrinkle in this new plan is that these items will be irrelevant when Mists of Pandaria drops. Obviously it's not a problem now, but it seems like it would save Dev time to design these quests around generic items (six flasks, six transmutes, six potions) in order to ensure that the quests are always viable for current content. Not a huge issue, but something they will likely have to revisit in the future (or not, clearly it took a few years for this update!). Of course if they did change it to a generic approach, it wouldn't be effective for the throttling effect they seem to be looking to achieve.
I hope this answers any questions you might have about how the new specialization quests work. If not, feel free to leave a note in the Comments and I will investigate further :)