Your search domain is initially based on your studio name. However, you can use whatever name you’d like. Use something similar to my_studio.search.snapizzi.com. There's no need to enter HTTP://www, but you MUST KEEP the .search.snapizzi.com as shown above in order for your URL to work properly. If you change the domain, any previously established links will always work.
You can also completely customize your URL by creating a new CNAME record for your domain and pointing it to your Snapizzi Search Page URL. Snapizzi does not have anything to do with setting this up. You must contact your domain provider for specific instructions on how to create and map an alias to your Search domain.