Your address
Your published site lives at a handle you choose, on the platform's own domain. You can change it in place from the bar at the top of the builder. A handle already taken by another business is refused, and so is a list of reserved words the platform keeps for itself — things like www, app, admin, help, support and status.
Renaming does not break the old address
This is the part worth knowing before you rename. The address you leave behind keeps working for a year: it still resolves to your business, redirects to the new one, and nobody else can claim it while that lasts.
So a card in somebody's wallet or a link in an old email keeps landing in the right place, and you are not forced to get the name perfect on the first attempt.
Your own domain
If you own a domain already, Settings → Custom domains walks it in two numbered steps: first prove the domain is yours by adding a TXT record it prints for you, then point the domain at us with the record it shows. Both records have a copy button, and the panel only ever shows a pointing record when there is a real target to show.
This is the Owner's to do — a Manager can read the panel but the controls are disabled. Where a deployment has not switched on live DNS checking, the panel says so in a banner up front and the check is refused rather than passing a domain that was never proven. Certificate state is reported as it actually is, so a domain waiting on one says it is waiting.
Putting booking on a site you already have
If you have a website elsewhere and do not want to move it, Settings → Website gives you a single line to copy. Paste it where you want the booking form to appear and it inserts your own booking page at that spot.
It works out which business it belongs to from the address of the script itself, so there is nothing to configure and nothing you can paste wrongly. Your site has to be published first. A booking made through it behaves exactly like one made on your own page: it is a request you confirm, and nobody is assigned to it automatically.