This comes up all the time so if you look through some previous posts you'll probably find the full code.  Roughly:
1) Add a function mouseClicked() - Processing will call this whenever a mouse button is clicked
2) In that function check whether mouseButton == LEFT or mouseButton == RIGHT and set a global variable called e.g. animate to true or false as appropriate.
3) Wrap either all your drawing code or just the bit where theta is adjusted in an if statement that only runs that code if animate == true.

Most of what you need is in the mouseClicked example.

Something like:

1. boolean stopped;
2. void mousePressed()
3. {
4.   if (stopped)
5.   {
6.     loop();
7.   }
8.   else
9.   {
10.     noLoop();
11.   }
12.   stopped = !stopped;
13. }

This is one way of doing it where I've just stopped it from updating theta.  The other is to either do all of the drawing or not based on the value of animate (which would be more efficient in this case).

1. import processing.opengl.*;
2. float theta = 0;
3. float dTheta = TWO_PI/360;
4. float radius = 200;
5. 
   
6. // If this is true the spheres will move
   
7. boolean animate = true;
8. 
   
9. //Lights, Camera, Action
10. void setup()
11. {
12.   size(800,600,OPENGL);
13.   noStroke();
14.   camera(width/2, height/2, width, width/2, height/2, 0,0, -1, 0);
15.   lights();
16.   fill(255);
17. }
18. 
    
19. void draw()
20. {
21.   background(255);
22.   float x,z;
23.   x = radius*cos(theta);
24.   z = radius*sin(theta);
25.   directionalLight(255, 255, 255, 0,0, -1);
26.   pushMatrix();
27.   translate(width/2+x, height/2, z);
28.   sphere(50);
29.   popMatrix();
30.   pushMatrix();
31.   translate(width/2-x, height/2, -z);
32.   sphere(50);
33.   popMatrix();
34.   pushMatrix();
35.   x= radius*cos(theta+PI/2);
36.   z= radius*sin(theta+PI/2);
37.   translate(width/2, height/2+x, -z);
38.   sphere(50);
39.   popMatrix();
40.  
41.   pushMatrix();
42.   // translate(width/2, height/2+x, z);
43.   translate(width/2, height/2-x, z);
44.   sphere(100);
45.   popMatrix();
46.  
47.   // Changing theta is what makes the spheres change position
48.   // If we don't change it the spheres stop moving
    
49.   if (animate) {
50.     theta = (theta + dTheta)% TWO_PI;
51.   }
52. }
53. 
    
54. void mouseClicked() {
55.   if (mouseButton == LEFT) {
56.     animate = true;
57.   } else if (mouseButton == RIGHT) {
58.     animate = false;
59.   }
    
60. }