Why does the color red always appear pixelated on TV and videos on the PC?
It's not an illusion - it's called chroma subsampling.

Most video codecs do not represent colour in full resolution. This allows for more efficient "lossy" compression because it takes advantage of the fact that the human eye is more sensitive to brightness ("luma") than colour ("chroma"). Most lossy codecs lower the chroma resolution to half or one quarter of the overall resolution, so you may only get one pixel's worth of colour for every four pixels of brightness. This dramatically decreases the amount of data needed, with only a small loss of apparent quality.

It's slightly more complicated though: the brightness is actually made up of the sum of the three colour components Red, Green and Blue. And they're not encoded as RGB, that would be require more bandwidth, they're encoded as YUV. Y corresponds roughly to the green component, and the U and V are Y minus the red component and Y minus the blue component (a gross approximation, actually - if you want the whole formula look here).

In most codecs the U and V components are sampled at a lower resolution than the Y. this is expressed in the three-way ratio you often see if you hang around video forums too much, e.g. 4:2:2 or 4:2:0. For a two row rectangle of pixels the numbers represent:


  "width of sample region (Y samples)":"UV samples in the first
  row":"additional UV samples in the second row"


A common example of this notation is in the codec name "proRes422" the 422 part of the name comes from 4:2:2 meaning for every a 4x2 rectangle there will be 4 Y samples in each row 2 UV samples in the first row (half the horizontal resolution) and 2 UV samples in the second row. So proRes422 has half the chroma resolution of the luma.

On the internet and on telly you're most likely seeing everything in a 4:2:0 codec. In every 4x2 rectangle of picture there are only two UV samples (the 0 means that there are no additional samples on the second row). So the colour part of the image is composed of chunks 2x2 pixels in size, in other words, one quarter the resolution.

This means the red channel on its own has one quarter the resolution of the overall picture. 

TL;DR the red looks pixelated - because it actually is.