Why is the sky blue?

The sky is blue because of light passing through it. Light from the sun enters Earth's atmosphere, the the layer of gases (air) that surrounds Earth, and hits the atoms and molecules in the air. The light from the sun is made of many colors of light that, when we look at them together, look white. 

Source: Hyperphysics

This is called scattering. Light is an electromagnetic wave that can come in many colors (see What is Color?). What we call "color" refers to the wavelength (and frequency) of light. The concept of wavelength is described in What is Color?. Certain wavelengths, or colors, of light are scattered more often than others. Blue light (shorter wavelength) is scattered more frequently red light (longer wavelength). This scattered light goes in all directions, including into your eye. 

Consequently, during the day, the sky appears blue since the scattered blue light, when you look at the sky not around the sun, you are seeing this scattered blue light. At dawn and dusk, you are looking more directly in the direction of the sun. Much of the blue light that the sun sent to earth is scattered in other directions, so less of the blue light reaches you. The light that reaches you tends to be orange and red. Consequently, the sky looks red or orange.

 You might now be wondering why blue light is scattered more than red light. This concept is a bit complicated. When light travels, it has an electric field and magnetic field that travel with it. An electric field is used to describe how charges act when they are at a certain position in space. A magnetic field describes how magnets act when they are at a certain position in space. Electric and magnetic fields are related by physics that I will not get into at this time (look up Maxwell's equations or Michael Faraday if you are interested). The electric field of light varies over the light's wavelength. When light of a given wavelength comes upon a particle, the light's electric field causes the electrons in the particle to move along with it. These electrons have charge. When charges move, they radiate light. Shorter wavelengths of light tend to interact more with the electrons of the particle, since the electric field of the light varies on a length scale similar to or smaller than the size scale of the particle. In a low-density gas like Earth's atmosphere, the charges give off (radiate) light in all directions and the energy of the light coming in is the same as that of the light coming out. This is called Rayleigh scattering. Many other types of scattering exist, some of which we use in our research. If you want to learn more, I encourage you to check out Hyperphysics.