Your PAWsite was published the moment you created it! However, for at least the first few weeks, until the site is indexed by search engines, no one will know about it unless you share it's URL aka link.
When you first created your PAWsite, you chose its link aka URL. You may have left the default suggested URL, which used your shelter or rescue name.
After you've created your PAWsite, you can see its link when you're logged in to edit your PAWsite at http://www.adoptapet.com/shelter/shelter-site/manage
For example, your link might look like this: