Why do some autofocus systems need a circular polarizer instead of a linear polarizer?

First we need to clear up a confusion caused by photographers too often not having paid attention in physics class. There are two types of polarization, linear and circular.

Linear polarization defines what plane the light is oriented in. It could be horizontal versus vertical, for example. Sky light bouncing off of a horizontal dielectric, like a puddle for example, will be horizontally polarized. This effect depends on angle, and there is actually a angle at which all the light is 100% polarized (all horizontal, none vertical). A linear polarizing filter oriented to pass only the vertical component (the dot product with the vertical direction) will block the horizontally polarized sky reflection off the puddle. With a good polarizer and with the light bouncing off the puddle at just the right angle, it will block almost all of it.

Circular polarization is a separate attribute of light than linear polariation. It is sortof the "handedness" of the light. Right-handed light is blocked by a left-handed circular polarizing filter, and vice versa. These types of polarizing filters are often used in 3D glasses that aren't the red-blue kind. This allows you to not have to have your head perfectly level for each eye to get its intended image. Left-handed light is still blocked by a right-handed polarizer regardless of how much it is turned.

That all made sense until the aforementioned photographers who didn't pay attention in physics class got envolved. As far as I have even seen, photographic polarizers are all of the linear type. This makes sense because they are intended for effects like enhancing or attenuating reflections off of puddles and the like. A circular polarizer wouldn't be useful for that. What is called a "circular polarizer" in photography, is not a circular polarizer as understood by the rest of the world.

So what are photographic "circular polarizers" and why do we have them? Originally, straight linear polarizers were used in photography. They did exactly what they were intended to do, which was messing with puddle reflections, scattered light from the sky, etc. Then along came cameras with built-in light meters. In a SLR, there is nothing between the lens and the film or sensor while the picture is being taken, so the linear polarizers are fine for that. However, to do metering before the exposure usually requires bouncing light off or a mirror. The light also needs to go to the viewfinder, so there are often partially-reflecting mirrors envolved to split the light. Unlike regular silvered mirrors where light bounces off a metal layer, partial mirrors usually envolve light bouncing off of a dielectric somewhere. That light is then linearly polarized. Or put another way, such a partial mirror splits the light in a different ratio depending on polarization angle. That's bad if the exposure meter thinks its seeing a known fraction of the light coming in.

The solution was to add another layer to the polarizing filter on the front of the lens. The front of the filter passes only the portion of the light aligned with whatever orientation you rotate the filter to. A second layer was added that you can think of scrambling the linear polarization. In more technical terms, this is referred to as a "1/4 wave plate", as pointed out in a comment by Mattdm. Now the partial mirror, which is in effect a somewhat polarizing filter, will pass about the same fraction of light in each direction regardless of how the linear polarizer at the front of the lens is rotated.

This situation is so common that "circular polarizers" (actually a linear polarizer with a de-polarizer after as discussed above) are now more common in photography than true pure linear polarizers. In a lot of cases, the "circular polarizers" will be more available and cost less due to volume.