Away messages are little notes you put on instant messenger services such as AIM and ICQ to tell people that you're not at your computer at the moment. Kind of like the Internet equivalent to the answering machine message.
The actual content of these messages varies widely, from program defaults such as AIM's "I am away from my computer right now.", to actual informative messages ("In Acapulco for the weekend, cheating on my wife"), to a few random lines specially chosen by the person who is not there.
This last type ususally tends to reflect the personality of the absentee, in much the same manner as putting a picture of yourself in front of your webcam does the same.
I'm not going to tell you what my away message is, because I have this alarming fear this could easily become a GTKY node...