The K Desktop Environment (KDE) is a free software project which aims to make a complete user-friendly desktop environment. The project was initially started in 1996.
The first versions of the K Desktop Environment were simply called KDE. The project has been trying to re-brand itself since KDE 4 was released in 2007. The modern KDE desktop environment is by KDE assets called KDE Plasma or just "Plasma". The same assets refer to the application libraries as the KDE Frameworks.
The KDE Plasma Desktop
The KDE Frameworks
The KDE application libraries are based on Qt, it's essentially a feature-adding layer above Qt. They makes it rather simple to write really advanced software in C++.
One big advantage to the KDE Frameworks over that foot desktop thing's libraries is that a program written using them works perfectly in all desktop environments. The KDE frameworks follow the Freedesktop standards closely. This ensures that a program written primarily with KDE in mind will work perfectly on Xfce, fluxbox, IceWM and other environments.