Facebook's not calling them "away messages," they're calling it your "sidebar status," but it's pretty much the same thing.
If you've forgotten what an away message was, here's a refresher: There was a time in the late '90s/early 2000s when the best/only reason to fire up the old dial-up connection was to chat with your friends on AIM.
You had a "screen name," probably something like Snoopdizzle420 or sugarayfan3938475 or xxXXxbabeegurlxXXxx (I'm still wondering what the Xs were for). When you were forced to actually walk away from your computer, you had to have the perfect away message to show how cool and witty you were.
Anyway, Facebook's version is being tested out in Australia and Taiwan. Users there can swipe left in the Messenger App for iOS and Android and bring up the sidebar, where friends' sidebar statuses will be displayed. Actual examples given by Facebook: "drinking an orange latte," and "watching the game at home."
There's even a spot for a little graphic you can choose to represent your activity, which is sort of more Myspace than AIM.
Everything is a remix.