How do objects reflect light to show different colors?

In the visible spectrum of light, all colors have difference frequencies of wave lengths. So, an object that is green would absorb all the frequencies of light with in the spectrum except green, the light reflected from the object then hits our retina and then our brain tells us to interpret that object to be green. The frequencies that are absorbed in to the object are then converted into energy. That's why when you wear black cloths, black absorbs all visible light, you feel warmer, because the light it absorbs is converted into heat. And white would the opposite from that.

If you are wondering why they would reflect that certain color, it is because all atoms and molecules have electrons. Electrons in atoms have a natural frequency that they vibrate when light hits them. The light with the same frequency as the natural frequency of the electrons in the atoms would be reflected.