You have it almost right...
The declaration is simpler than it looks: you declare an array of images (PImage[]), then you reserve the slots in that array (new PImage[lenght]), but these images are empty  (null).
So you populate it in the loop, loading the images and assigning their reference in the array.
Of course, you need to vary the name, otherwise you end with n times the same image in the array...
That's why usually the images are numbered, so that you load the image 003 on the third iteration.
But there is another way, if you don't want to rename your files.
Another issue is in the draw() loop: you get a random number (a float) and try to assign it to images[i], but i doesn't exists there, and you can't assign a float in a PImage variable.
What you want looks like:

1. void draw() {
2.   background(255);
3.   // Array indexes start at 0!
4.   float r = random(0, numPuddles); // Get a random number between 0 and the number - 0.00001
5.   int ir = int(r); // Transform to an integer between 0 and number - 1
6.   PImage img = images[ir]; // Get the corresponding PImage
7.   image(img, 0, 0); // Show it
8. }

I decomposed each step to be clearer, usually you find instead something like:

1. void draw() {
2.   background(255);
3.   image(images[int(random(0, numPuddles)), 0, 0);
4. }

How to load images of various names: put their names in an array too!

1. // Special syntax to initialize an array at declaration time
2. // Put 6 names there!
3. String[] imageNames = { "puddle-red.png", "puddle-green.png", "puddle-purple.png" };
4. PImage[] images = new PImage[imageNames.length];

Then in the loading loop:

1.   for (int i=0; i < imageNames.length; i++){
2.     String imageName = imageNames[i];
3.     images[i] = loadImage(imageName);
4.   }

You forgot a } to close the one opening after setup(). The one before draw() is closing the one after for (). If you put the caret after a { or a } you can see the matching one to be highlighted.

Hey thanks for that tip, its really helpful. 

Prefer imageNames.length, as you are sure it is accurate: if you add one image name, it will be still correct, while you can forget to update numPuddles...
Glad you sorted out the issue yourself.

scttcrawford, you have to make a loop over your array, and test the name against each entry of the array (using .equals() of course). Once found, break or return the index.
Alternatively, if you feel adventurous, you can use a HashMap<String, PImage> linking each image name to the pre-loaded image.