No offense but from what I've seen from you in the forums, it clearly stands out that you don't really have the basics down yet.
Read up on linear algebra and hone your math skills, because honestly we can't answer all the maths questions that would ever come to your mind (also it would be faster if you would just read a good book). With that being said here's an answer to your question:
A vector is just pretty much a point in space (in 2D space it's likely to have 2 components: X and Y, in 3D space it has X, Y and Z).
If you think about vectors as arrows or as a line than you are most likely wrong.
A vector points from A to B, but if we're talking about a single vector than we "imagine" the other one to be at (0, 0[, 0...]).
You can make a vector that points from A to B just by subtracting A from B.
You can subtract two vectors by subtracting all of their matching components one-by-one (sorry if my explanation is not clear but English is not my native language either).
Vector3f a = new Vector3f(3f, 5f, 2f);
Vector3f b = new Vector3f(12f, 6f, 5f);
Vector3f aToB = new Vector3f(b.x-a.x, b.y-a.y, b.z-a.z);
LWJGL's vector class also has a subtract method if I'm not mistaken but I was just trying to demonstrate the principle here.
We call difference X = DX, and difference Y = DY, so basically you just have to subtract the A vector's X and Y components from the B vectors X and Y components, resulting in DX and DY accordingly.
You can get the angle between to points using the method that CopyableCougar4 wrote:
double angleInRadians = Math.atan2(dy, dx);
Please note that the method's first parameter is DY and not DX as one would expect.
Also note that this method's return value is not degrees but radians.
If you want to convert radians to degrees you can do the following:
double radians = 0.6;
double angles = radians*180.0/Math.PI;
Size in Cougar's code should be called distance or length, that's the distance between the two vectors or as often referred to, the length of the vectors (btw I think this is wrongly called so, because calculating the distance between two vectors and a single vector is different).
The distance formula is this:
Keep in mind that I'm not a mathematician and there might be something wrong with my explanation or in my code, although I was trying to keep my post error-free.
Hopefully this cleared things up, but as I said in the beginning of my post, you should pick up a book on the subject.