Time Machine uses a unique ID (
UUID), which is specific to a volume (i.e. a partition of a disk), to associate the volume with its backup. In OS X releases before 10.7 Lion, when people exchanged their disk or migrated to a different Mac, this feature has made it difficult to get Time Machine to continue adding to the backup history of the previously used disk.
In OS X Lion and now also OS X 10.8 Mountain Lion, there is a nifty new Terminal command called
tmutil, which makes this whole drama a breeze.