Hello,

You have to problems of algorithmic.

First, you reset the image at each call of the function "drawBlueSquare", so you need placed the function "background" on the "draw" function.

Secondly, the variables x and y aren't reset at each image. After each loop, you need to reset the variable.

But there an other problem. i search ...

And sorry for my english

Charlie :)

You call background() in drawBlueSquare, so you erase any previously displayed square!
Move this call to the start of draw().
And you must reset x on each row, and y at the end of draw(). Oh, and x must skip every other square, so double the increment...
And each other row, you have to shift the starting offset.

Some changes, based on my remarks:

1. void drawBlueSquare(int x, int y) {
2.   noStroke();
3.   fill(0, 0, 255);
4.   rect(x, y, 8, 8);
5. }
6.  
7. void drawRowofSquares(int y) {
8.   while (x <= width) {
9.     drawBlueSquare(x, y);
10.     x += 16; // Nice shortcut
11.   }
12. }
13.  
14. void draw() {
15.   background(255, 0, 0);
16.   y = 0;
17.   while (y <= height) {
18.     if (y % 16 == 0)
19.       x = 0;
20.     else
21.       x = 8;
22.     drawRowofSquares(y);
23.     y += 8;
24.   }
25. }

for the shift at each row :

if( (x/8+y/8)% 2 == 0) {drawBlueSquare(x,y);}

It's the same graphic from a very recent post -> http://forum.processing.org/topic/how-to-recreate
Gonna re-publish my example over there here: 

// http://forum.processing.org/topic/how-to-recreate

size(500, 500);
noLoop();
smooth(4);
noStroke();

background(#F00020);
fill(#0000F0);

final int dim = 10, dd = dim << 1;

for (int row = 0; row < height; row += dim)
  for (int col = (row/dim & 1) == 0 ? -dd : -dim; col < width; 
  rect(col += dd, row, dim, dim));

Very understandable for a beginner, GoToLoop...

Note: the original code uses while loops, but each of them need an init, a stop condition and an increment, so it makes more sense to use for loops, where everything is grouped in the same place, avoiding errors like the lack of proper init:

1. void drawBlueSquare(int x, int y) {
2.   noStroke();
3.   fill(0, 0, 255);
4.   rect(x, y, 8, 8);
5. }
6.  
7. void drawRowOfSquares(int startX, int y) {
8.   for (int x = startX; x <= width; x += 16) {
9.     drawBlueSquare(x, y);
10.   }
11. }
12.  
13. void draw() {
14.   background(255, 0, 0);
15.   for (int y = 0; y <= height; y += 8) {
16.     // if y is multiple of 16 (on even rows), startX is zero, otherwise it is 8
17.     int startX = y % 16 == 0 ? 0 : 8;
18.     drawRowOfSquares(startX, y);
19.   }
20. }