This is one applet for constructing Fibonacci Squares and the Fibonacci spiral. The Fibonacci spiral is made up of quarter-circular arcs whose radii are consecutive Fibonacci numbers.
You can read detailed instructions under the applet, short instructions at the left side of the applet, or see video instructions by following the link below. The order of steps is important.
Video Instructions: http://screencast.com/t/0Czgk3OU2o

1. Click on the Quarter Circle button.
2. Click on the green square to create a square under it (the color will change).
3. Drag the green square to align the blue point with the free end of the quarter circle.
4. Rotate the green square by dragging the red point. We want to align the red side of the green square with the radius of the quarter circle.
5. Click the Increase button until the side of the square equals the sum of the sides of the previous two squares.
6. Repeat the procedure from step 1.