An anti favor is created when one of your friends doesn't do something for you that could have easily been done. You then say...
I owe you an anti favor.
You will now not do something for them that could easily be done by you, but may cause discomfort for he/she to do.
Example will explain situation:
Me: Can you get me a cup too?
Friend:....."gets one cup for himself)
Me: Why didn't you get me a cup?
Me: That is one anti favor.
...I then say I am getting everyone a beer and do not get my friend one. That ladies and gents is an anti favor.