A Twitch Extension is a website displayed in an iframe on Twitch. As @BarryCarlyon mentioned, it is a collection of files that make up your “website”. Your website can be a simple static content site that just shows “hello world”, or it can be something complex that requires an Extension Back-end Service (EBS). The EBS is optional and is not hosted on Twitch.
During testing, you specify where your front-end files are located, which could be local or on the server that’s hosting your EBS. But after you’re done testing, you upload your “zipped” front-end files to Twitch. Now your front-end is hosted on Twitch, and if you have an EBS, your front-end will still communicate to it by AJAX. Just make sure the paths to your EBS are correct in your front-end AJAX and CORS is set properly on your EBS server.