In your case, to make it work under IE, you need to put your app domain and SharePoint domain into the IE local Intranet zone:
Also make sure that automatic authentication is enabled:
Please, take a note, that this approach will work only if the user is inside organization domain, for outside the user will still be prompted for credentials. If you want to implement fully SSO, then you need to go with ADFS, this approach requires ADFS configuration inside SharePoint and changes in your app in order to support ADFS SSO.
For other browsers that might be harder. You can try to search for "integrated windows authentication" plugins for Chrome or Firefox.
IE settings can be pushed to the users via domain polices, but for other browsers every user need to manually download and install plugins.