a database, depending on the underlying technology, is basically a collection of files that represent the data contained therein.

A migration usually involves copying those file(s) to a new server - so first the new server (installation of underlying database software) would need to be set up, the files copied over, attached and loaded into the new server software, and then all database references in software will need to be repointed to the new server address (or DNS entries changed, etc.)

that's a very quick explanation, that's rather technology agnostic - obviously it can vary due to a number of reasons but this is the basics of it.