Is is possible to recover a database from the 'raw' files (.mdf and .ldf) of the old server to a new SQL Server instance on a new server?
Please pay attention. The answer from Julien is correct. BUT if you attach the database to an SQL Server 2008 instance it will be upgraded! If the application can't handle SQL Server 2008 and needs SQL Server 2005 you're in trouble. Take an extra step and make a backup.

Furthermore you have to create the needed logins on the new server. You might want to attach the old master under a new name for looking up the details.