Cloning does to a volume something like what fork does to a Unix process. The original volume and the clone share the same disk space, which in the clone is read-only, and in the original volume is copy-on-write. The original and its clone are rather like two identical collections of hard links to the same files. When you change a file in the volume, the link to the read-only copy of that file is broken, and a new file is created. So the clone is like a backup of a volume at the time it was taken. It requires very little disk space unless the volume changes substantially.
The clone of a volume can itself serve as a limited backup of the volume. In one of our cells, we clone all user volumes every night, and if you accidentally delete or mangle a file, you can read or copy the file from the clone without bothering the system administrator. When you backup a Coda system, you don't need to stop using it; instead, you clone each volume to be backed up, and you dump the clone to tape, or to a staging disk from which you can do something else with it.
Cloning was used to produce read-only replicated volumes, but Coda seems to be dropping support for these.