Achievement

Achievements will be a future update coming to Minecraft. Speaking to Gamasutra, Notch said: “I like achievements. I know a lot of people don’t, but I like them. I’ve had the idea to make achievements kind of like the in-game questing. So you’d be able to see the first achievement in a tree of achievements, and you unlock the top ones first before you can unlock the ones further down."

He also said: “So the first one might be to chop down a tree, or kill a chicken, and then these branch into more things you can do. Hopefully it would encourage people to try new areas. It could converge into a big task, like kill a dragon or something, which would put a kind of narrative into the achievement tree.”

Finally, asked if such a move would risk leading players down a preset path, rather than encouraging exploration and invention as Minecraft does in its current state, Notch said: “Definitely. I’d want these achievements to feel like things you can try, rather than these are things you have to do. People can follow them, but only if they want to.”

The Beta 1.4 update was originally intended to include achievements and statistics, however implementation was pulled from the release because it was not working yet.

March 18th Update
On March 18th, Notch talked about Achievements and Statistics on his Blog. He said the following:

"I worked on the achievement and stats system. One design issue was dealing with offline mode and syncing the achievements once you get a connection again, but that’s been solved. We’ve got a couple of people in the office who don’t like achievements in games at all, so the goal is to design something that they’re fine with."

Also, he added that Achievements will not be chores:

"Achievements will NOT be chores like “cut down 10000 trees”, but rather challenges like “ride a pig off a cliff”. Stats, however, will be used to keep track of how many trees you have cut down. The long term plan is to show achievements and stats from the profile page on minecraft.net as well, in case you want to brag."