As new versions of SQL Server are released, certain features or ways of doing things change. So something that worked in an older version of SQL Server may not work in a newer version or may behave differently. The compatibility level is a way of telling SQL Server to use the older methods of doing things and is applied at a database level. So if the compatibility level is 90 then it will use all the 2005 options, if it is 80 it will allow 2000 features that have been removed from 2005. The compatibility option does not affect the version of the database - so a 2005 database cannot be restored into 2000 as it is still a 2005 database even if the compatibility level is 80. Also, it does not stop you using features that are above the compatibility level you have set - so if you were to set the level to 8 then 2005 features would still be available to you.

As for changing the level after moving a database, it is advisable to keep it at the current level until you have tested the application with the higher level.