Sunlight is white. Or close enough to white that calling it anything else would require more precision than is useful here. It contains all the visible wavelengths mixed together — what you see when you pass it through a prism and get a rainbow. So when sunlight hits the atmosphere, why does only blue reach you from overhead, while reds and oranges arrive at the horizon near sunset?
What scattering means
Scattering, in optics, is what happens when a wave of light encounters a particle and gets redirected in a different direction. This happens constantly in the atmosphere. Air is mostly nitrogen and oxygen molecules, and those molecules scatter light as it passes through them.
The important detail is that they do not scatter all wavelengths equally. Shorter wavelengths — the blue and violet end of the spectrum — are scattered much more strongly than longer wavelengths like red and orange. The relationship is not linear: blue light is scattered roughly ten times more than red light. This is what Rayleigh scattering describes, named after the physicist who worked it out in the late nineteenth century.
Why blue and not violet
Here is where it gets slightly awkward. Violet light is actually scattered more than blue light, because it has a shorter wavelength. So by the logic above, the sky should look violet. The reason it does not has two parts.
First, the sun emits less violet light than blue to begin with — the spectrum is not uniform across wavelengths. Second, and more significantly, the human eye is less sensitive to violet than to blue. The cone cells in your retina that respond to short wavelengths peak around blue frequencies, not violet. The sky is not blue because violet is absent; it is blue because that is what your visual system makes of the mixture of blue and violet reaching your eye.
The sky looks the colour your eye is most sensitive to among the wavelengths that scatter most. A different eye would see a different sky.
Why sunsets change the calculation
At sunset, the sun sits near the horizon and the light reaching you has to travel through a much longer path through the atmosphere than when the sun is directly overhead. The exact distance through the atmosphere at a high sun angle is around eight kilometres. At sunset, that same light may travel through eighty kilometres or more of atmosphere before reaching your eye.
Over that longer path, most of the blue light gets scattered away from the direct line between you and the sun. It is redirected in every direction and most of it ends up going somewhere other than your eye. What is left — what has survived the long journey through all that scattering — is the red and orange end of the spectrum, which scatters weakly and mostly travels in a straight line.
This is also why the sky near the horizon at sunset can be quite pale or whitish even as the sun itself looks deeply orange: the horizon sky contains light arriving at intermediate angles, a mix of wavelengths that has not been as cleanly filtered as the direct solar disk.
Why clouds are white
Clouds are made of water droplets, which are much larger than atmospheric gas molecules. Particles much larger than the wavelength of light scatter all wavelengths roughly equally — this is called Mie scattering, and it does not have the colour-selective property that Rayleigh scattering does. All wavelengths are redirected similarly, and the result is white. Storm clouds look grey or dark simply because the droplets are dense enough that very little light passes through them at all.