To get started in the Achivements system, click this little shield button you'll find in your action bar. You may also type /achievements, and a hotkey can be bound to it from the keybinding interface.
Here's the main screen of the Achivements UI. It shows your Achievement Points total, how many achievements you've completed, and recent updates to achievements and statistics.
These are some examples of General achievements. Achievement Points for each achievement are noted, as well as the date on which that achievement was completed.
This shows the detail on one particular Achievement, the one to reach level 70. Here you can see all the prerequisite achievements, and how many points they were worth -- the one to reach level 60 is being moused over; I assume the rest are levels 10, 20, 30, 40, and 50. This confirms, by the way, that achievements are awarded retroactively, as this character was already level 70 when he was transferred to the beta realms, and yet is awarded the achievements for reaching levels 10 through 60.
This shows the detail view for an achievement that has a progress bar and isn't part of a chain. It shows that seven out of seven bank slots have been purchased, thus unlocking the achievement.
This shows the detail for "The Right Stuff", illuminating that earning the previous riding skills are pre-requisites.
This shows some Cooking achievements. I must unlock The Cake Is Not a Lie! But I don't have that recipe yet. Maybe it will remain a lie after all. (By the way, "NYI," as seen on The Epicurean, stands for "Not Yet Implemented.")
Cropped out of this image is that this screenshot (as well as the two following) are from the "Statistics" tab of the "Statistics and Achievements" UI. That's right, the game now tracks all sorts of stats about your character, as seen on the left-hand side. Here, for instance, we could see how many bandages or potions we've used.
Here the interested player can see how many times he's been ressed, and by what classes. My "times considering resurrected but logged instead" would be something like three, not counting disconnects; it just feels untidy to me to log out while dead.
This lists all sorts of counts about the various travel methods that we face in the World of Warcraft. However, I am not looking forward to watching a city full of people trying to get their "Blood elf spins in a row" score up. Oh well, we all face minor inconveniences. I hope you've enjoyed this tour through the Achievements UI. For everything Wrath of the Lich King, keep reading WoW Insider!