I'm surprised that after a few years in that major, you haven't had any of those elements covered for you, but here goes:
Library: just a collection of code, usually centered around a specific functionality. An engine is a library, a framework is a library, etc.
Wrapper class: "Wraps" a class or piece of functionality inside another class, usually for ease of calling the thing being wrapped in the wrapping class' language. Ex: SharpDX (a graphics library and API) has C# wrapper classes around the C++ DirectX classes. (this is grossly simplified for ease of explanation)
Framework: A library (ha!) or set of libraries that provide a lot of basic functionality that will be used to build upon. XNA is considered a framework because it doesn't actually give you a game, just the toolkit to make one.
Game Engine: is a framework. If you take a completely coded game, and remove all the stuff that specifically makes that individual game, what you're left with is the engine: all the code that makes everything work but that knows absolutely nothing about the specific assets it's loading, dialogue that happens in a game, level layouts, etc. Game engines often (unless you're some AAA studio like Epic or Id) are a result of writing a specific game, and then keeping all the code that can be re-used to make another game.
GDK: usually a framework (or engine) coupled with a few handy editor tools (level editor, asset packager, etc)
API: Application Programming Interface - You can think of them as tools to put in your coding toolbox. Graphics APIs, networking APIs, Physics APIs...basically a black box of code in a library with (hopefully) documented interfaces for using that functionality. Or not black-boxed in the case of open source. API does a thing, you need something that does that thing, you use that API in your project.
For the specifics:
Unity3D: Game Engine/GDK/Framework/APIs. This spans a lot. It has its own engine, you can program with its APIs, it has editors (that you can extend in code)
UDK: same as Unity3D
Allegro: as per its wikipedia entry, it's a library that contains multiple APIs. You could also consider it a framework (are you catching that some of these are loose definitions that overlap a lot?). It has everything you would need to make an engine, but is not one itself.
DirectX, OpenGL, SDLnet: APIs. Well, libraries of APIs if you stick to the concept of an API as a singular tool.
Qt Creator: had to look this one up, I guess it's just an IDE? (Integrated Development Environment, think Visual Studio, Eclipse, Netbeans, etc)