What is the easiest way to determine the date on which I received a specific badge?
Use the "recent activity" data, accessed with the little orange (or grey) envelope at the top of the page, then switch to the "badges" tab. It lists the day you received each badge. (You can adjust the exact date range by tweaking the URL, so as to hone in on a specific time period you are interested in.)
What voyager said, and then the data dump if you really want to know.
Excerpt from "anatomy of a data dump":
+--- badges.xml | - Id | - UserId, e.g.: "420" | - Name, e.g.: "Teacher" | - Date, e.g.: "2008-09-15T08:55:03.923"
The new interface makes it a lot easier, just go to your Profile -> Recent Activity -> Badges.
If it was recently awarded, and depending on your activity, you can see them at your profile. Otherwise I guess you are out of luck.