Failover is the process of shifting I/O and its processes from a primary location to a secondary disaster recovery (DR) location. This typically involves using a vendor's tool or a third-party tool of some type that can temporarily halt I/O, and restart it from a remote location.
This will temporarily halt I/O, suspending data copying and mirroring activity that may be going on from the primary location to the secondary location. This will then bring applications and I/O up from that remote location.
During activity at the remote site, changes are usually tracked so that their original location can be re-synchronized and restored to service by just replicating the data between the start and end of the DR event back to the primary location when it comes back up. Failback is the process of re-synchronizing that data back to the primary location, halting I/O and application activity once again and cutting back over to the original location.