White light entering a prism is actually made up of many different kinds of light. Lights of different colors have different "wavelengths" (this is the distance between the peaks of two waves: see here about waves).

Our eyes sense the different wavelengths and we see this as color. When white light enters the prism, each kind of light within white light (all the colors) are reflected at different angles because of the different wavelengths. This effectively separates all of the colors present in white light into different bands which we can see as the spectrum. (prism monochrometer) What's really cool is that by using two prisms, if you have two prisms, you can use one to separate the light into the colors, and another to put it all back together into white light!

This is a great question! When white light passes through a prism, the prism disperses the light into what is called its constituent colors. Prisms are able to do this when they are transparent and have flat surfaces with some angle between them. When light travel through the air hits the prism, it changes its speed, and is thus enters the prism at a new angle (it's "refracted"). This angle depends on the refractive index of the material in the prism, which can vary with wavelength of light. Wavelength of light in turn, corresponds to different colors of light. So these different colors will travel at different angles in the prism and exit as a "rainbow" or spectrum of colors, rather than as white light.

The process happens because of the way that waves work. Light travels at different speeds depending on what it's shining through. In a empty space, light travels at about 300,000 kilometers/second, which is the fastest speed, but in a substance like glass or water it moves slower. Also, different colors of light move at different speeds in different materials, and this movement at different materials causes the light waves to separate out into colors, which is why you see rainbows. Prisms work just like rainbows, except that the substance is glass instead of drop of water.

"White" light from the sun is made from the combination of many colours of light. When light interacts with matter, the strength of the interaction depends on the colour (the energy) of the light. So when white light (of many colours) enters a prism, the different components interact with different strengths.

What this means is that blue light will be bent differently than green light, than red light, and so on. This causes the many colours of light to separate, and is why you see a rainbow of colour. This happens in a prism, but also happens very notably in water droplets, and this is what leads to rainbows in the sky.