SQL Server compatibility level
When databases are upgraded either using backup restore method or via attach/detach process, the compatibility level of the databases do not get changed automatically, and show as old comp. level. However this is not a major problem, but point to note is that you may not be able to use various features , that are available on newly upgraded SQL server. 

As mentioned by owen, you should refer the MSDN doc to refer the difference between the Compatibility-level setting of 100 or lower *with* 
Compatibility-level setting of at least 110 .

Also,  running an upgrade advisor  will help you in identifying for any issues while you do that upgrade or change in compatibility level.

If you have the test environment you can always test the scenario before making any changes in prod and once you are good to go, just make sure you have the latest backup of system databases on production server. In case anything goes wrong, you can recover or rollback from the backups.

Also, an additional read here Bad habits : Clinging to old compatibility levels