what is the correct syntax for the creation of arrays?

I've read this: http://unity3d.com/support/documentation/ScriptReference/Array.html

But can't figure out when they're making up words to describe what they're doing or when they're using commands or reserved words.

Is there a perfect, clear definition of all the ways to create/declare arrays floating around somewhere? Real Reference material perhaps?

By way of example:

// This appears to create an array

function Start () {
    var arr = new Array ();

// but it looks all the world like declaring a function.
// why does it look like declaring a function?
// Why would you use this method when there are ways to declare it more structurally that don't confuse?    

// This also creates an array    
var values : float[];
// apprently this is a "builtin" array.  How do I know that?
// How do I know when an array is a builtin array or not?
// float is a type, right?  How many different types are there that arrays can be?
// how many different types are there that builtin arrays can be?

Well the problem here is that javascript accept a lot of thing without complaining, so you don't really understand what's happening. Let's take a look at your code :

var arr = new Array();

This is just a short a way to declare an Array. In fact, here is what happened, but was hidden :

var arr : Array; // Declaration
arr = new Array(); // Initialisation, or instanciation. It's the same as i = 0, but Array is an object and need a call to it's constructor. 

If it's a little fuzzy for you, you need to learn a bit more about "programmation orient objet" (don't know the good english term, sorry)

About the difference between classical array and built-in array, it's about performance. The first provide some useful tools (resizable, you can put anything in, etc ) but the second one is quicker. And you can build one with any type you want, even one you created, you just can't put different type in the same built-in array. Well, maybe that's not very clear ^^. I should let the code talk for me !

// Here is a class I created. "A" become a type, just like float.
class A
{
   function A(){ toto = 0; } //Constructor
   var toto : int;
}

function Start()
{
   // This is an instance of A, in a simple variable
   var simpleA : A = new A();

   // This is a classical array, I put anything I want in it.
   var classicalArray : Array = new Array();
   classicalArray.Add(simpleA );
   classicalArray.Add("Hello");
   classicalArray.Add(5);
   // Now you have in classicalArray => (simpleA, "Hello", 5 )

   // This is a built in array of A. I fill only with A variables
   var builtinArray : A[];
   builtinArray = new A[3]; // This create the array of size 3
   //I fixed this line -Peter G

   builtinArray[0] = new A(); // This create a new A
   builtinArray[1] = new A(); // This create a new A
   builtinArray[2] = new A(); // This create a new A
   // Now you have in builtinArray=> (A, A, A )
}

I hope it's more clear !

Is there a perfect, clear definition of all the ways to create/declare arrays floating around somewhere? Real Reference material perhaps?

I think you really should pick up a book and learn programming. All of this will become immediately clear after finishing a beginners book in about any language.

I guess there are mainly two ways arrays (or rather collections) are formed. One way to do it is to use built in arrays, and another way to do it is using links/references to next item (linked lists or any node based architecture).

Fundamentally about every other collection class internally use either, or a combination of the above mentioned methods. Those collection classes include Array, List (and many other classes).

You were a bit confused about the constructor of Array.

But it looks all the world like declaring a function.

Why does it look like declaring a function?

What you're doing with new Array() is allocate memory for the type Array and invoke its constructor. The constructor is a function that is called when you are creating new instances of classes. This is similar to Awake() in your scripting. Again, I'd strongly advice you to go pick up a book about programming.