Usually one sets the MX records on the old host to point to the new host. The trick with doing this during a move is, you can't use the domain name for the new host, you have to use the actual hostname of the server, which you can find by looking up the IP for your domain name, then looking up that IP to see what it returns.
For instance, if your domain name is yourdomainname.com you would look up yourdomainname.com and find it has an ip of 172.16.32.56.
You would then look up that IP (172.16.32.56) and see that has a hostname of server53.com.
Then you'd set your MX record on the old host to point to server53.com.
With "IMAP transfer", you can connect to both hosts simultaneously (you may have to use the server name for the old host) and you can then drag and drop from one server to the other. Not at all an official term or definition, hence why you didn't find it via Google, but nevertheless a handy trick.