Yes. When you click "Publish", there will be instructions in Site Designer for specifying your URL before it goes live. By default, you will be assigned a *.houzz.com subdomain based on your Houzz Username (ex: http://wilsonkandb.houzz.com). If you want to use a custom domain (i.e. http://www.example.com) you need to own or purchase one first through a domain registration service (ex. GoDaddy, Namecheap). You cannot register a domain through Houzz.
If you decide to specify a custom domain for your site, you will need to modify the DNS records of that domain to point to our servers for your site to work.