If you currently have a site hosted elsewhere you will need to have a copy of those files from that site, so backup all files on your old host's server to your local computer.
Log into your ( old) site via FTP and download all the files. These files will be images, logos, HTML and CSS files.
If you use another program/script on your current ( old ) site, make sure you get a copy of the database and any files from this too. Programs/scripts you might be using can include a forum, special contact forms or upload scripts..etc.
