Paper or any suitable drawing surface in which to stick pins or the like.
2 Pins or other tools to stick to the surface, around which string can loop.
Put the 2 pins into the paper, thus determining the ellipse's foci.
Tie the string into a loop and lay it loosely around the pins. (Alternatively you can tie it to both pins, but this does not work as smoothly.) The more string you leave hanging, the larger your ellipse will be.
Put the pencil into the loop and push against it to make a triangle. Keep the string taut as you draw a smooth elliptical curve running all the way around the two pins.
Calculate and mark the positions of the foci of your ellipse.
Push a pin directly into both of the foci.
Calculate and mark any point on the circumference of the ellipse.
Tie a string to one pin, loop it around the other, and use the pencil or a third pin to loop it around the predetermined point on the circumference. Loop the string back around the pin it is tied to, and then tie it tightly to itself. Untie the string from the pin, leaving the string sitting loosely in a single loop around the two pins.
Draw the ellipse, keeping the string taut as you press the pencil against the paper and the inside edge of the string.