Can I restore a SQL Server 2008 R2 backup to SQL Server 2008 R2 Express?

Yes.

It's the database version that matters and is what gets attached to the database, not the edition of MSSQL used. As long as the database version is the same, you can move it between the free (Express) and paid versions to your heart's content... not that it's a process I'd recommend for recreational purposes.

EDIT: Since it's been mentioned, the relevant database-limitation in SQL Express 2008 R2 is a size limit of 10GB. It also has some limitations on the services and resources available to MSSQL Server, but those won't prevent you from restoring the database to it, only (possibly) using it how you'd like.

Full Technet feature comparison for 2008 R2 versions linked here.

It depends. If your database includes features that are not supported by the edition you are restoring to the restore will fail. For example, enterprise edition features like data compression and transparent encryption. This is in addition to the physical resource limitations that exist in Express.

If your database falls within the supported features / limitations of Express you should be all set.