Active database files are constantly changing. Therefore, trying to simply make a copy of an active database file will not result in a valid backup file, but will generally result in a corrupted file; the data written at the start of the file will not necessarily be consistent with the data written later in the file.
So, most database systems have their own internal backup methods, to ensure that a backup file accurately represents the data as of a given moment in time. This backup would be atomic in the database sense - even if it takes hours to write out the file, the contents will be consistent with when the backup started, as though no changes were made to the file for all that time. However, changes are still happening to the file (hence, the online part of the description).
The only other way to get a consistent database backup file is to shut down the database. If the database is not active (not online, in other words), then no changes are happening, and the OS can copy the file without risk of corruption. This would be an offline backup.
Note that there are other differences between these two backup methods. An offline backup is normally going to be a simple copy of the database files; in theory, restoring the database would be as simple as replacing the current files with the backup files. An online atomic backup is usually stored in a different format, and has to be restored by running commands in the DBMS.
