You CANNOT do this - you cannot attach/detach or backup/restore a database from a newer version of SQL Server down to an older version - the internal file structures are just too different to support backwards compatibility. This is still true in SQL Server 2014 - you cannot restore a 2014 backup on anything other than another 2014 box (or something newer).
You can either get around this problem by
using the same version of SQL Server on all your machines - then you can easily backup/restore databases between instances
otherwise you can create the database scripts for both structure (tables, view, stored procedures etc.) and for contents (the actual data contained in the tables) either in SQL Server Management Studio (Tasks > Generate Scripts) or using a third-party tool
or you can use a third-party tool like Red-Gate's SQL Compare and SQL Data Compare to do "diffing" between your source and target, generate update scripts from those differences, and then execute those scripts on the target platform; this works across different SQL Server versions.
The compatibility mode setting just controls what T-SQL features are available to you - which can help to prevent accidentally using new features not available in other servers. But it does NOT change the internal file format for the .mdf files - this is NOT a solution for that particular problem - there is no solution for restoring a backup from a newer version of SQL Server on an older instance.