What are these balls of wool bound together with a rubber band?
These are just pom poms on elastic  bands. Those particular ones appear to be made from rabbit fur. That is, according to the less that useful description given from the site where this image came from. 


  Korea cute fashion real rabbit fur ball hair ring hair rope single ball wholesale


 Page appears to be in Chinese Simplified and that is what I got from Google Translate

The ones pictured are used as decoration or as an accessory. 

Coincidentally we had another user ask about how to make those recently. Have a look at the answers to this question and you should get an idea of what to do. The only thing that is different is the presence of the elastic band. If you do want that it would just be a matter of attaching that into the pom pom as it is being created.