When you look at some part of the sky away from the sun, the light you see from the sky is sunlight that has been scattered toward your eyes by air molecules. The sunlight induces wiggling dipoles in the molecules, causing them to emit light in directions different from the original direction of the light.
The molecules of air scatter higher-frequency light waves more strongly than they do lower frequencies. To your eyes and brain, this makes the scattered light look blue rather than white.
(A previous version of this page claimed that it was also responsible for the color of the Earth seen from space, but Phil Plait argues that the dominant effect there is the greater absorption of long wavelengths by the ocean.)
This scattering also makes the sunset appear red or orange.
Pictures taken on Mars by various robot probes show a sky in shades of orange, pink and tan. This is because the atmosphere of Mars is very thin and dusty. Scattering is dominated not by gas molecules but by reddish dust particles.