On a desktop computer, "app" is short for application, and refers to a fully developed program such as Pro Tools, Studio One or Ableton Live.
On a mobile device such as a smart phone or tablet, an "app" is a program that functions within that device. These used to be a sprite application that would respond quickly on the device, much like a messaging application. But now that phones & tablets have become sophisticated, the "apps" are just as fully developed as their desktop versions (as seen by their file sizes).
The term "plug-in" was originally developed by Adobe Systems developers who wanted to provide extended services to Photoshop (e.g., an effects filter). These software programs ran "inside" Photoshop and became so popular that Adobe provided plug-in capabilities in all their desktop programs.
Plug-ins do not stand alone like applications do. Plug-ins run inside applications.
In the audio realm, a plug-in is usually an effect that is placed in the signal chain of an audio channel. Compression, EQ, reverb, etc., are considered plug-ins.
All modern DAWs (digital audio workstations) designed for recording, mixing & mastering music support third-party plug-ins.