What is an anti-aliasing (AA) filter on a camera sensor, and do you want one?

Short answert:

The AA filter is a filter that sits on the sensor and blur the image by a little bit, this makes the image less sharp but helps prevent all sort of problems - the most noticeable is moire patterns, the filter is less needed in higher resolution cameras for still pictures but is usually still required for video.

Long answer (simplified so I don't have to give the math background):

You can't represent anything that is smaller than 2 pixels in the image - you just don't have a way to represent this, a single small object in the image will effect the color of the pixel that "contains" it.

Repeated object that are two small will cause repeated pixels to change creating a pattern that isn't in the scene you are photographing.

When something (the scene) looks like something else (moire pattern) is is called "aliasing".

The Anti-Aliasing (AA) filter splits the light so that everything is blurred over the length of two pixels (or the area of 4, 2 horizontal X 2 vertical) hiding the small features the camera can't capture and preventing the unwanted patterns.

The AA filter is also called a "low pass" filter, the repeating object are an "high frequency signal" and a low pass filter blocks the high frequencies passing only the low frequencies.

The higher the resolution the smaller the objects have to be to be less than approx two pixels so the AA filter isn't needed as much on high resolution cameras.

HD video is just 2 mega pixels, so even on an high resolution camera video is still low resolution and most of the time (depending on the sensor design and in-camera processing) is still prone to aliasing and requires an AA filter.