Low-pass filters have long been used on camera sensors to prevent moiré or interference effects when you photograph subjects with fine, regular patterns. You may have seen this effect on old TV sets with presenters wearing striped shirts or check jackets – those horizontal or vertical lines clash with the rectangular array of photosites on the camera sensor.
Today's sensors have much higher resolutions, so it takes a much finer pattern to produce this moiré effect, but in theory it could still happen when photographing woven fabrics, for example, or fine rectangular patterns in man-made objects.
It's not just the rectangular grid of photosites that's the problem, but the color filter array that's placed over them to capture full-color images. The most common layout is the 'bayer' array, with a regular pattern of two green photosites, one red and one blue in a repeating 2 x 2 grid. This can cause color fringes and artefacts if the camera captures ultra-fine lines that don't straddle all four photosites; it's rare, but with a super-sharp lens and the 'wrong' subject, it can happen.
What does a low-pass filter do?
The solution is the low-pass filter. The technical description is a filter which allows lower frequencies to pass through (coarser detail, in this case) but attenuates or blocks higher frequencies (finer detail). In a nutshell, a low-pass filter slightly blurs the pixel-level detail to reduce or prevent moiré effects. This slight softening can be disguised with a little extra image sharpening later.
For a long time, low-pass filters were taken for granted as a necessary part of sensor design, but then Nikon started launching cameras like the D7100 and D800E with no low-pass filter. These produced visibly sharper fine detail (if you looked very hard), and Nikon's hunch that the pixel pitch of the newest high-resolution sensors was so fine that moiré wouldn't be a problem any more was proved right. Now, it's common for makers to tout the absence of a low-pass filter as a selling point.
So we don't need them any more?
That doesn't mean moiré can't happen. Some software tools have moiré suppression options for images where it does appear, and Pentax cameras have a particularly novel approach, with an 'anti-aliasing filter simulator'. Pentax DSLR sensors don't have a low-pass filter, but they can use their sensor-shift Shake Reduction system to apply tiny high-frequency sensor movements during the exposure to simulate the effect of a low-pass filter. The idea is that you use the filter simulator only for those shots where you think moiré might be an issue.
When Nikon launched the D800 it came with a low-pass filter over the sensor, but Nikon also launched a D800E variant with no filter. And when it launched that camera's successor, the D810, Nikon ditched the low-pass filter entirely, apparently with no ill effects.
Canon, meanwhile, offers two versions of its EOS 5DS. The regular model has a low-pass filter over the sensor, while the EOS 5DS R has a low-pass 'cancellation' filter. It's not clear how that works, but the intention is to provide a second, even higher-resolution model for those photographers prepared to take their chances with moiré.
Sigma's Foveon sensors might produce lower-resolution images (in pixels) than regular cameras, but they nevertheless capture uncannily sharp fine detail – partly because there's no color filter array on the sensor and it doesn't need a low-pass filter.
Fujifilm, meanwhile, doesn't use low-pass filters for its X-Trans sensors, claiming that the X-Trans design's unique color filter array is sufficiently 'random' to eliminate the risk of moiré effects.
It looks like low-pass filters are no longer essential in modern camera design, and taking them away does in theory improve very fine detail rendition, even if it takes a very sharp lens and perfect shooting technique to show it.