Original User: RupertEssinger
Hi Kathy
There are three parts to your question:
Apps: All of the Story Map application templates can be downloaded and installed on your own web server or website. In addition, some of the apps are also available hosted in Esri's ArcGIS cloud (i.e. the Map Tour, Swipe, Basic, and Text and Legend apps). The Shortlist app that you mentioned is only available as a download. So you host it yourself and you can customize or tweak the code as much as you like.
Images: Images referenced by the Story Maps applications can be anywhere on the web. You can put your images on your own web server or website, in a file/photo sharing site like Flickr, or stored in ArcGIS Online. For example when you make a Map Tour you can choose whether you want to reference your images via URLs to where they live on the web, or import them into ArcGIS Online (where they get stored in a feature service as attachments). For the Shortlist app, the easiest way to handle images is to put them en masse on your own web server or website and then just reference them via a URL.
Web map: The web map (or web maps in the case of Story Map apps like Text and Legend that show multiple maps) that is referenced by your app is hosted in Esri's ArcGIS cloud. The layers the web map(s) are either stored in the ArcGIS cloud or can point at services being served up from other servers, like ArcGIS Server. So the web map is not installed on your own server (but it can contain data that you or others may be serving via ArcGIS Server). (If you are in a large enterprise using ArcGIS Server you also have the option to use the Portal For ArcGIS extension to make your own private cloud for hosting web maps and data).
But the short answer in terms of using the Shortlist app is that the app itself and your images can be on your own server, but the web map the Shortlist displays is in the cloud.