What does it mean when a PS4 game says PS4 PRO ENHANCED on the cover?

Solution 1:

"PS4 Pro Enhanced" simply means that the game makes use of the improved performance that the PS4 Pro offers. In other words: the game offers improvements when played on the Pro.

What those improvements are depends on the game. Most of the time, games will offer improved performance or graphical quality.

Horizon Zero Dawn for instance, lets players choose between either improved resolution or improved performance. Nier Automata on the other hand increases resolution to Full HD (1080p) and manages to keep framerates more consistent at the same time.

Games that are not "Pro Enhanced" do not offer any improvements when played on the Pro. Persona 5 for instance renders exactly the same on PS4 and PS4 Pro; same framerate, same resolution, same graphics quality. Please note that Pro Enhancement can be patched in (e.g. Uncharted 4 and Infamous), so even if it doesn't say "PS4 Pro Enhanced" on the cover, the game can still offer improvements after installing recent updates.

For games that are not "enhanced", the PS4 Pro also offers a "Boost Mode" which simply improves performance for games (if possible) in exchange for possibly causing issues.