Also note that the compatibility level of a database is not necessarily the same as the version of SQL Server that it is hosted in (please forgive formatting errors in what follows).
Level Version Can be executed in:
80 SQL Server 2000 SQL Server 2008 through SQL Server 2008 R2
90 SQL Server 2005 SQL Server 2008 through SQL Server 2012
100 SQL Server 2008 and SQL Server 2008 R2 SQL Server 2008 through SQL Server 2014
110 SQL Server 2012 SQL Server 2012 through SQL Server 2014
120 SQL Server 2014 SQL Server 2014 through SQL Server 2014
So you could be running 80 compatibility in SQL 2008, but when you restore to SQL 2012, the compatibility level will be automatically upgraded to 90 (the minimum compatibility level supported). It will do this "silently". Upgrading the compatibility level can involve "breaking" changes which could affect your applications, and should be thoroughly tested.