Apps is an abbreviation for applications. An application is a piece of software, that executes a job or fulfills a certain purpose. An application may be public (Google, Youtube, Spotify, ...) or privately hosted (selfhosted applications). Homarr requires application to have an internal and external URL.
The external URL is the link to an application. If you click on an application on your dashboard, Homarr will open said application.
The internal URL defines, how Homarr should communicate to the application. The external URL can be set differently than the internal one, if one of your applications is not accessible outside a specific network (eg. not exposed).
A widget will display data or enable you to control the state of a system. Some widgets use the integrations to display a set of data or give you control over the applications (eg. disable ad blocking).
An integration is the back bone of apps and widgets. It acts as the bridge between them. Integrations can be configured inside applications (Edit application > Integration tab). Widgets will then search for applications that are set to the appropriate integration (eg. Ad summary widget will use all apps that are set to the PiHole and AdGuard Home integration).