You have to create a global variable to store the current shape of the object.
Since it's either a square or a circle, a boolean data-type is enough -> boolean squareOrCircle;

final static byte  shapeSize  = 30;
final static color shapeColor = 30;
final static color bg = #FFCC00;
boolean squareOrCircle;

void setup() {
  size(300, 300);
  noLoop();
  noCursor();
  fill(shapeColor);
  
  rectMode(CENTER);
  ellipseMode(CENTER);
}

void draw() {
  background(bg);

  if (squareOrCircle)    rect(mouseX, mouseY, shapeSize, shapeSize);
  else                ellipse(mouseX, mouseY, shapeSize, shapeSize);
}

void mouseMoved() {
  squareOrCircle = !squareOrCircle;
  redraw();
}

great, thank you! what does redraw(); do? i noticed that without it the shape doesnt change. 

Since your original code didn't use draw() anyways, even though draw() is necessary for dynamic drawing animation...
I chose the half-way -> draw() turned off by noLoop();

By default, draw() is automatically called by Processing 60 times/second, that is, 60 FPS.

But since this code just needs to redraw the animated object only after a user moves the mouse...
There's no need to keep erasing and redrawing it so many times each second.

That's what redraw(); does -> allows draw() to be called once!

Of course, you can remove both noLoop() + redraw(), and the animation works normally.
Only this time, draw() is called @ 60 FPS instead of each mouse movement!
frameRate() sets how many FPS though.

very helpful information, thanks!