I would like LWJGL3 but will go with whatever I can get.  My PC specs are:

Dell Win-Vista SP2 64-bit
Model Inspiron 530s
IntelCore 2duo cpu
DirectX11
NVIDIA GeForce GTX 750 Ti

Can my desktop run LWJGL

Why don't you try it yourself?
If you have at least a Java 1.6 JRE then it should work.
It also depends on what exactly you mean by "LWJGL".
If, for example, you want to use the OpenGL binding of LWJGL to create an OpenGL application with an OpenGL context version supported by your graphics card/driver, then it should work.
You should just give it a try: Download LWJGL from lwjgl.org/download, get a JDK or use an IDE with integrated compiler and try build and start the getting started demo at lwjgl.org/guide.

LWJGL is a library. You use it to write your own programs. I am not quite sure what you think "working" means in that context. LWJGL just offers you a java binding for typical C libraries. (plus a little bit of fluff)

Okay, thx.  Having a Vista OS puts me in a hot seat where I have to double check everything.  Vista now means every other program I download fails either today, or tomorrow during an update, because MS are butheads and Gates can stuff his OEM disks where sun shines not.

I have Eclipse and Netbeans.  What I am doing is learning java programming and ending up on some "lets build a game" tutorials on YouTube where they argue over what version of LWJGL and OPENGL they should use.  Back when I studied Blender I discovered that because I use Nvidia I can render everything thru my graphic card setting it to GPU compute.  Java programming is all new to me.

In a nut shell, I'm just a student learning everything for the first time. 

In this case LWJGL is definitely not the right tool for you. It is very low level programming and has little to do with java. Its supposed to give you C functionality and exposes a minimal interface for graphics programming as well as some additional support libraries. If you are new to java (or programming in general) you will have a very difficult time learning openGL straight up. I would suggest you better stick to the AWT or libGDX or other high level libraries if you really want to create a game.

Why is every java game building tutorialist demanding we have it?  Why do they have it?

Having what? LWJGL?
I am sure that not every such tutorial is using LWJGL. I have seen plenty which use AWT or libGDX or other libraries.