If you are building a telescope or if you want to see if a telescope primary mirror has a good 'figure' before you buy it, you need to perform an optical test of the mirror.

A properly figured primary Newtonian mirror is a near-perfect parabola. How close to perfect does it need to be? A good telescope mirror has less than 1/4 wavelength of light error on its surface! This means that the surface imperfections should be smaller than 100 nm. This is a very small margin for error but amateurs telescope makers can achieve this and better if they are persistent and careful.

In order to test the figure of the primary mirror we must build the following:
Mirror mount,
Optical tester,
Optical mask

Then I'll describe how to use these tools to determine the figure.

Step 1: Theory of Operation

The 100 nm accuracy of the surface requirement is a very tight tolerance and can not be measured directly using such things as rulers or depth gauges. These tight tolerances can indeed be achieved using normal hand tools by magnifying the surface angles. We can find the focus point of the mirror by shining a narrow slit of light onto the mirror and observing the shadows as a knife edge is passed in front of the returning light.

Let us assume that we have a perfectly shaped spherical mirror and our knife edge is near the focus point of the mirror.

See Figure A.
Looking down from the top, if we illuminate the mirror at the focus point the light will hit the mirror and return exactly to the focus point.

See Figure B.
If we move a knife edge inside the focus and on the right, the knife edge blocks the light returned from the right side of the mirror and does not affect the light from the left side of the mirror.

See Figure C.
If we move the knife edge outside the focus and on the right, the knife edge blocks the light returned from the LEFT side of the mirror.

So, when the knife edge is inside the focus, moving the knife edge left and right makes a shadow move across the mirror in the same direction as the movement of the knife edge.
When knife edge is outside the focus, moving the knife edge left and right makes a shadow move across the mirror going in the opposite direction of the knife edge.
When the knife edge is exactly at the focus point and the knife edge is moved from left and right, the mirror dims uniformly, i.e. the shadow doesn't appear to move left or right but the entire mirror dims.

A good telescope mirror does not have a spherical shape but a parabolic shape. For a parabola, the outer areas of the mirror focuses farther away from the inner areas of the mirror. We can therefor use this knowledge to determine the precise shape of the mirror by finding the focus points of each area of the mirror.

Step 2: Tools and Materials

Tools:
Table Saw with dado blade,
wood chisel
screwdriver

Materials:
(Note: all of my dimensions are in US units.  Metric equivalents are perfectly fine.)
For Mirror Mount:
    3 foot by 1.5 foot plywood, 3/4 inch thick
    2 foot by 1.5 foot plywood, 3/4 inch thick
    2 6 inch angle brackets
    1 L hook
    12 screws 3/4 inch long for the L brackets
    3 flat head wood screws for 'feet'

For Optical Tester:
    6 inch wide by 32 inch long by 3/4 inch thick hardwood board
    3 single sided utility blades
    2 magnets 1/2 inch diameter
    1 depth gauge with 1 inch or more of travel with readings every .001 inch
    Small bright bulb or LED (I used a krypton bulb)
    6 inch 3/32 inch thick or better metal angle iron
    6 inch 3/32 inch think flat metal iron, 1 inch wide
    2 1/4 inch threaded rod, 20 threads per inch, 3 inch long
    2 knobs that fit 1/4 inch threaded rod
    4 springs
    miscellaneous wood screws and glue

For Couder Mask:
    1 light-weight cardboard large enough to cover the mirror being tested.

Step 3: Building the Mirror Mount

A mirror mount is needed to hold the mirror during testing.

If the mirror is a 'thin' mirror (i.e. the thickness is much less than 1/8th its diameter) then the mount must hold the mirror without deforming it. The weight of the glass can deform a thin mirror.

In my case, I was making a full-thickness mirror, which is a mirror whose thickness is 1/8th the diameter or more, so mirror deformation was not an issue.

If mirror deformation is a factor in your use, you may want to replace the two wood blocks I used with a leather strap which distributes the weight of the mirror across the entire bottom half of the mirror.

The mirror mount I made is shown in the attached pictures.

Step 4: Build the Optical Tester

The optical tester is a device which shines a slit of light onto the mirror and finds the focus point of each area of the mirror.

It is essentially an X-Y table with a 1/1000th of an inch accuracy in the X dimension.

First Construct the base,
Construct the middle layer,
Construct the top layer and assemble,
Add the light source and the depth gauge.

Step 5: Construct the Bottom, Middle, and Top Sliding Boards.

Pictures are worth a thousand words so I hope you can tell what I did form the pictures because my words are probably a little confusing.

I used three 6 inch by 7 inch, by 3/4 inch pieces of wood, one for the bottom plate, one for the middle plate and one for the top plate.

Using a dado blade in my table saw I carefully cut a slot into the board just wide enough to hold a wood slat and deep enough so that slightly less than 1/2 the depth of the wood slat is protruding when placed in the slot.

I then cut a similar slot in the middle board. When the slat is inserted into the base slot, the slat will lay inside the slot in the middle board and allow the boards to slide in only one axis.

On the other face of the middle board I cut a similar slot that is rotated 90 degrees from the first slot.

On the top board I cut a similar slot so that a slat placed into the middle board's slot and the top board's slot allow the top board to move perpendicular to how the middle board moves.

Place two slats in and glue them only to the middle board this allowing the boards to slide.

Step 6: Complete the Base

Complete the base by adding three 'feet' so the base sits very stable.

Add an arm that holds the light slit.

Add a metal plate with a 20 threads per inch bolt so that the middle and top plates move when the bolt is turned.

Add a .001 inch dial gage to very accurately tell how much the top and middle plates have moved.

Add a spring to hold the middle plate against the threaded bolt.

Step 7: Complete the Middle Plate

Add a plate attached to the middle plate that hold a 20 threads per inch bolt that presses against the top plate moving it perpendicular to the other bolt.

Add an arm with a utility blade to the top plate so that when the bolt is turned the blade 'cuts' the light returning from the mirror.

Add a spring so the top plate is pressed against the bolt.

Step 8: Using the Device.

Place the mirror being tested in the holder and the optical tester at twice the mirror's nominal focal point.

Shine a slit of light from the tester to the mirror and back again. Proper positioning is critical.

As mentioned before, when the top plate is moved the utility blade cuts the light returning from the mirror.
When moving the knife edge from left to right, if we see the mirror darken from left to right also then the knife edge is inside focus.
If the darkening moves opposite the knife edge, then the knife edge is outside focus.

Since we want the mirror to have a perfect parabolic shape, we measure precisely the focal points of each area of the mirror.
To help us focus on areas of the mirror, we make a couder mask.

A couder mask has little windows cut out that expose sections of the mirror. For example, the couder mask will have windows for the outer 1/2 inch of the mirror. Another set of windows expose the next inner 3/4 inch of mirror. Ideally as the windows expose mirror surfaces closer to the center the window gets larger. This is because we want each window to represent equal amounts of light entering the eyepiece, and at the outer edges the area is larger due to the larger circles of the window.

To build a couder mask we start by deciding the outermost dimensions of R1 and R2. We then compute R3 so that the circles exposed represent equal areas of the mirror. We continue for R4, R5... and so on until we reach the center of the mirror or so. The centers of newtonian mirrors are in the shadow of the secondary mirror and they really don't contribute much light.

Step 9: Optical Testing

I couldn't find any good videos of the mirror testing. I have the following video which just shows how the shadows move over the surface. As you can see this mirror has a rather large hole in its center. If you use the mask, you can watch the shadows and record their positions to the nearest .001 inch. You then compare the actual results with the perfect parabola. If you are building your mirror you then change the polishing strokes and retest allowing time for the mirror temperature to stabilize, keeping good notes of the results given your polishing strokes. You can then keep polishing using the techniques that move the mirror surface closer to a parabola.