What happens when one of the filegroup is full within SQL server
Every table or index has at least one partition.  Each partition is stored on a single filegroup*.  So when that filegroup is full and none of its files can be expanded, any operation that needs to allocate additional space for a partition that is stored on that filegroup will fail.

*Except that a table can be configured to store its blob data on a separate database filegroup, or filesream filegroup.