This is a really interesting question. First of all we need to think about light. White light is actually made up of all the colors of the rainbow. Light takes the form of waves and all the different colors have a different wavelength.
Red light has the longest wavelength (720 nm) while violet has the shortest (380 nm). We see objects as different colors depending on how they interact with light waves - a red flower is absorbing all the colors except for red. The red wavelengths are are reflected so we see the flower as red. So now that we have that sorted, what does it have to do with the sky being blue?
The air around us isn't just empty, it is mainly made up of nitrogen and oxygen gases. The molecules of these gases are too small to see but big enough to alter the path of light waves. (Note: lots of people will tell you that it is dust and pollen in the air that scatters the light but this is not really the case). If there were no gases in the air, all of the wavelengths of light from the sun would travel in a straight line to our eyes and we would see it the sun as white against a completely black sky. Instead, the molecules in the air get in the way of the light and start bouncing it around all over the place. However, the air molecules aren't large enough to effect the red light with the longest wavelength. On the other hand, the shorter wavelengths (violet and blue) hit the molecules in the air and are scattered in all directions. The scattered light hits other air molecules and the result is that, by the time it reaches our eyes, the short wavelength light is coming from all directions.
Violet light is affected the most by this scattering so why do we see the sky as blue and not violet? Well the receptors in our eyes that detect color are for red, blue and green light, and other colors are seen due to differing stimulation of these three receptors. This means that we are far more sensitive to blue light than violet, so the sky looks blue.
At sunset, we often see the sun and the surrounding area as orangey red. This is because the light from the sun has much further to travel to reach us than during the day when the sun is directly overhead. The blue light becomes so scattered that it doesn't reach us. The red is also scattered to some degree but still gets to our eyes. So the sky looks red.
You can show this light scattering for yourself with a bowl of water with a little soap or milk added to it. If you shine a white light through the liquid the beam will appear slightly blue from the sides as the blue light is scattered by molecules in the water and will be reddish at the end as this light remains unaffected.
