When I bought my polarizer, a friend told me that I should get a circular one, because the linear ones can mess with the autofocus. Is this true? What should each be used for?
A polarizer works in the way that it will let through only the light that is polarized in the same direction as the filter is currently turned. It is true that many AF systems have problems with this. To solve this, circular polarizers have a layer behind the linear polarizing filter that "re-polarizes" the polarized light in a different way so that the AF can function properly. The linear polarizer does not have this extra layer. The wikipedia article explains this quite well, I think.
As far as I know, there are no cases when you would want a linear polarizer instead of a circular one within the context of placing it on a camera lens. On any camera that a linear polarizer will do the job, the circular one will work just as well, but not the other way around.
Yes, it's true - sometimes. Some AF systems work OK with a linear polarizer under some or all conditions, and others fail all the time. You'd have to try it and see.
When would you want a linear polarizer?
- They are often cheaper.
- Since there's only one polarizing layer, not two, light transmission is often greater.
Because many AF and metering systems have problems with polarized light, circular polarizer has an added layer which "re-polarizes" the light by transforming the the polarization of the light wave into a sort of helix shape. Therefore you should only get circular polarizers in usual cases.
The exception is when you want to use two polarizers to create a tunable ND filter. In that case, the front polarizer has to be linear, and then you can control the darkening effect by adjusting relative position of the two polarizers.
A 'circular' polarizer is just a linear polarizer with a quarter-wave plate behind it that repolarizes the light in a circular kind of way. But the polarized light removed by passing through the linear polarizer is gone, and thus is not reintroduced by the "re-polarization" of the remaining light.
A quarter-wave plate is not another linear polarizer. It is arranged 45° with respect to the linear axis of the polarizer. As the polarizer is rotated to alter the effect of the filter with regard to light from a particular direction, the quarter-wave plate is attached (bonded) to the linear polarizer and rotated as well. The quarter-wave plate is always turned 45° with respect to the polarizer's linear axis.
The polarization of the light by the quarter-wave plate allows phase detection autofocus (PDAF) systems and light meters to function properly. PDAF systems often fail when trying to focus light that has passed through a linear polarizer alone. Light meters placed past the viewscreen in reflex¹ cameras are also affected by linearly polarized light.
There is no effect of the quarter-wave plate on the appearance of the photo compared to a linear polarizer without a quarter-wave plate. Neither film nor digital cameras record in which direction light is polarized when it strikes the sensor or film. The linear polarizer allows light polarized in one direction to pass through it. The light that is not allowed to pass through is gone. The quarter-wave plate can only act on the light that has been allowed to pass through the linear polarizer in front of it. It can't recreate the light that the linear polarizer blocked. So the light striking the film or sensor is the exact same light either with or without a quarter-wave plate behind the linear polarizer (other than the minimal amount of light lost due to transmission through a refractive medium as happens with every lens element in the optical path).
Two linear polarizer filters arranged 90° out-of-phase with one another will (theoretically) block all light from passing through (with or without a quarter-wave plate behind the second one). So-called 'Variable Neutral Density' filters are actually two stacked polarizers. As one is rotated with respect to the other, the varying angle between them blocks more or less of the light striking them.
¹ 'Reflex' is the 'R' in SLR and DSLR and refers to the reflex mirror used to divert light from the lens to the viewfinder.
Circular polarisers have an additional layer creating "unpolarised" light from the polorised light the first layer admits. If you have any kind of beam splitter (like an optical viewfinder) or phase detector autofocus afterwards or TTL sensor working from light reflected back from film, this may make a difference. With most digital cameras these days, it doesn't. This may change when the usual contrast-based autofocus systems these days get replaced or supplanted by phase-based ones on higher-end cameras.