To start, Processing's auto format-er is your friend. Go to Edit > Auto Format to clean up your code a bit...

I have rarely seen a while loop used properly. They have their time and place, but you should almost never use them to draw. The draw() function already loops for you, and the screen is only updated at the end of draw. In this case, you must define your variables at the global level (so outside of draw(), along with ellipsexposition) and then update them once every frame. Also, you will want to add checks to wrap the values around after they exceed their limits (either 0 or 255). Note that the current setup will not give you smooth color transitions (by my standards, at least...), but it is a step in the right direction.

Adapted Code:

1. //Global color variables
   
2. int colorr=0;
3. int colorg=0;
4. int colorb=0;
5. 
   
6. float ellipsexposition=0;
7. 
   
8. void setup() {
9.   size(500, 500);
10.   smooth();
11. }
12. 
    
13. void draw() {
14.   background(255);
15.   int endx=width/6*5;
16.  
17.   int ellipseyposition=height/2;
18.  
    
19.   //Change the variables in draw() (this was already here...)
    
20.   colorr=colorr-1;
21.   colorg=colorg+1;
22.   colorb=colorb+1;
23.  
    
24.   //... and wrap the values around at the ends
    
25.   if(colorr < 0)
26.     colorr = 255;
27.  
28.   if(colorg > 255)
29.     colorg = 0;
30.  
31.   if(colorb > 255)
32.     colorb = 0;
33.  
34.   fill(colorr, colorg, colorb);
35.   ellipse(ellipsexposition, ellipseyposition, 50, 50);
36.  
37.   ellipsexposition=ellipsexposition+=1;
38.  
39.   if (ellipsexposition<endx) {
40.     ellipsexposition=ellipsexposition+1;
41.   } else {
42.     ellipsexposition=0;
43.   }
44.  
    
45.   //No while loop
    
46. }

A concept sometimes hard to grasp in Processing is that
 a screen frame is only rendered at the end of draw().

We don't see a drawing happening at the same time graphics functions are called.
Rather, we see their final result for each draw() completion.

Rather than having a loop block which will render the whole animation,
we should make draw() the loop itself, since by default it's called @ 60 FPS anyways.

@Calsign Ah, I see. That makes much more sense. I originally started programming in basic, which is why a while loop was the first solution that came into my head when I was presented with this problem. Processing is somewhat different and it has been an adjustment thus far. Thanks for your time and help.

@GotoLoop Thank you, your input really complimented Calsign's explanation

http://wiki.processing.org/w/Technical_FAQ
Second article (and the others too!)