The process of a domain transfer from one supplier (provider) to another is called a change of provider. With a change of provider, you become a customer of the new provider and manage your transferred domain there.
The procedure is also known as a transfer of registrar, switch, or ChProv. A change of provider is not to be confused with domain transfer, which can take place from one contract into another within the same provider.
With a change of provider, only the domain itself is moved, not the data linked to the domain, such as your website or email accounts. Therefore, it is highly recommended that you back up your data before performing change of provider, in case you later need to upload it to your new provider.