What changes SQL Server Compatibility Level changes?
A compatibility upgrade changes the way the SQL Engine works and has the database level improvements/changes from the other versions.  This doesn't mean you can't use features outside of that though.  For example having a DB in a older compatibilty mode doesn't mean you can't use mirroring or AGs.  

It does mean the query optimizer, newer system provided functions, and other features will not work the same way or be available in some cases.  You will want to ensure your code operates the same way but your DR/HA and operational solutions shouldn't have changed.  

Run your code through the Upgrade Advisor and also read up on the breaking changes.  Read up on best practices for upgrades as well, Thomas LaRock has a good article on it as an example.