The release is likely missing a library/plugin or the library is in the wrong directory and or from the wrong directory.
Qt intended answer: Use windeployqt. see last paragraph for explanation
Manual answer:
Create a folder named "platforms" in the same directory as your application.exe file. Copy and paste the qwindows.dll, found in the /bin of whichever compiler you used to release your application, into the "platforms" folder. Like magic it works. If the .dll is not there check plugins/platforms/ ( with plugins/ being in the same directory as bin/ ) <-- PfunnyGuy's comment.
It seems like a common issue is that the .dll was taken from the wrong compiler bin. Be sure to copy your the qwindows.dll from the same compiler as the one used to release your app.
Qt comes with platform console applications that will add all dependencies (including ones like qwindows.dll and libEGL.dll) into the folder of your deployed executable. This is the intended way to deploy your application, so you do not miss any libraries (which is the main issue with all of these answers). The application for windows is called windeployqt. There is likely a deployment console app for each OS.