Transferring a website means transferring the files and any databases from a server / provider to another server / provider. This is not the same as transferring your domain name.
The process of transferring a website varies per website. In general, it is done by downloading files from the old server / provider and uploading them to the new server / provider. This is often done using FTP software.
Transferring a website by only transferring the files, is often not enough. Many websites often also have a MySQL database, which must be moved as well. This is often done via phpMyAdmin. After transferring a MySQL database, you often have to change the configuration files of your script, this differs per script. Information about this is available in the documentation of your script.
Some articles that may help you with FTP and phpMyAdmin are: