Message signing is a way you can prove you control a particular address, and demonstrate you have ownership of the funds.
This feature is currently only available for your Imported Addresses, but will be implemented for all addresses in the future. To sign a message, go to Settings -> Addresses, and scroll down. Choose which address you want to sign, and click More Options -> Sign Message.
A small window will display allowing you to provide a unique message or signature that identifies you as the owner.