ndmpcopy -- The ndmpcopy command enables you to transfer file system data between storage systems that support NDMP v3 or v4. The ndmpcopy command functions as a simple NDMP data management application (backup application) that performs data transfers by initiating a backup operation on the source storage system and a recovery operation on the destination storage system. The command establishes control connections to the storage systems and facilitates data connection establishment. After connections are established, it facilitates data transfer. You can use host names or IPv4 addresses of source and destination storage systems in the ndmpcopy command. For more details refer https://library.netapp.com/ecmdocs/ECMP1196992/html/GUID-0D32E86D-3DDC-4E91-AAA5-4C6DF05A4AB1.html
Vol copy --
You can copy one volume to another volume, by using the vol copy command. Enter the following command on either the source or destination system: vol copy start [-S | -s snapshot_name] source_volume dest_volume The -S and -s arguments specify the Snapshot copies to copy. source_volume and dest_volume are the names of the source and destination volumes. If a volume is on a different storage system, precede the volume name with the system name and a colon. For examples illustrating how to specify volume names, see “Examples of the vol copy start command” in the following link, https://library.netapp.com/ecmdocs/ECMP1368826/html/GUID-3C1E8471-8D47-4B0C-965C-1F6318DA2803.html
In addition to the descriptions of the move commands, you also asked about snapvault and snapmirror.
so they are designed to do differnet things, there will be some underlying technical sharing of process, but from a user perspective snapmirror and snapvault don't do what the other processes do.
both mirror and vault are methods for replicating data and subsequent block level changes to alternate locations.
the easiest way to think about them is mirror is a DR technology, whatever you have in prodcution mirrored to an secondary location, this includes all prodcution data and snapshots, so if you have 30 days snaps in produciton alongside your active data, you have exact same in Secondary, the secondary cannot hold different data.
This is where Vault comes in, SnapVault is NetApp to NetApp backup really, it take a copy of the active system at an alterante location and can maintain independent snapshots, so if you want to retain 7 days in production for quick recovery but maintain a year on a remote location for retention, SnapVault is the answer.
Lastly you asked about other move methods, well snapmirror is certainly a way of moving data between aggregates and controllers, but also vol move, this is non-disruptiove method for moving data between aggergates... there are some limitations to what can be vol-moved, but that is the neatest way of carrying out the move of volumes between locations.