Erik,
It sounds like either the application or the services in the application are secured. If these are hosted services, I would make sure they're shared with everyone and not just the organization. In regards to the security of the application, I'm assuming you have an IIS web server in which case you can go into IIS manager and check what type of authentication is being required for the app; you'd want to set that to anonymous authentication.
If the log in prompt is for ArcGIS Online specifically, I'd guess that your services aren't shared with everyone, you may also want to verify that anonymous access to your organization is allowed under Edit Settings > Security.
Hope this helps,
- Daniel