During a domain registration transfer, nothing changes except who the registrar is, and that doesn't really affect anything on the Internet.
If your domain registration expires while you're trying to transfer it over, then your site may go down though! We recommend always attempting a domain registration transfer at least a month before your domain is going to expire. There's no reason to even wait that long though, as you can transfer them at any time.
When you transfer a domain registration, it will also renew for another year.
Last updated: Dec 13, 2002.