The purpose of the secondary mirror is to deflect the light reflected by the main mirror out of the path of the incoming light. It is called the 'secondary' because it is the second optical surface in the light path.
Note A : Usually it deflects the light by 90 degrees. Any other angle would work just as well, in the rest of this page we will assume a deflection of 90 degrees.
Note B : Strictly speaking, a secondary is only necessary for a main mirror that is not a sphere. (A spherical mirror can be used for off-axis systems)
The secondary mirror has an oval shape, this is because the cone of light from the main mirror is intercepted at 45 degrees. The shape of the resulting figure is an ellipse. A Secondary is classified by its smallest diameter, the short axis (minor axis) of the ellipse.
The size of the secondary depends on the following factors; Focal length, Main mirror diameter, Distance of focal point to the optical axis of the main mirror, Area around the focal point which should receive as much light as the focal point itself. The formula is given below.
Example : Main mirror diameter : 200 mm, Focal length : 1200 mm (f/6), Distance between main mirror optical axis and focal point : 150 mm, Radius of area which should receive maximum possible light : 10 mm (This area would be smaller if the scope were for visual use only). The secondary thus needs a short axis of 42.5 mm.
The highest resolution can only be achieved in the circle of full illumination.
Since the secondary also stops light from reaching the main mirror, it is the cause of the most unwanted property of a Newtonian telescope, the obstruction. This obstruction has the effect of reducing the maximum obtainable contrast. The obstruction is normally expressed in diameter percentage, but sometimes also as an area percentage (probably because this yields a 'better' number). In the example of point 3, the obstruction is 21 %.
Scopes that are exclusively used for visual observations should aim for an obstruction of 20 % or less.
Scopes for imaging usually have obstructions above 20 %.
Experienced ATMers often warn against a reckless effort to reduce the central obstruction. It is much easier to 'damage' the scope by a too small secondary as with a too large secondary.
In order to get a fully symmetrical illuminated area around the focal point, it is necessary to apply a small offset to the center of the secondary. This is because the secondary is placed at 45 degrees with respect to the optical axis of the primary. That point of the secondary which is closed to the primary intersects with the light cone at a greater distance from the optical axis as the point furthest away from the primary. The offset is thus away from the focuser.
When the offset is applied, the secondary is no longer centered above the primary. This can be seen in the inside/outside focus images. It also might have an impact on the in-focus images, though as of yet I have not seen proof for this. (Please mail me if you know more about this)
Depending on the collimation method you use, you might have to take the offset into account.
For focal ratios of f/6 or slower the offset is so small that it should be ignored (or cannot be applied). For f/4 and f/5 it can be applied, and for faster mirrors (when used for imaging) it should be applied.
If the scope is used for visual use only, the offset can be ignored. (Assuming that one doesn't build ultra fast telescopes for visual use anyhow)
According to Texereau, the minimal acceptable surface accuracy for the secondary mirror is 1/8 wavelength. It is generally believed that the cheaper commercial scopes do not meet this requirement (or even get close).