In a nutshell, a solar cell works as follows: A special material is constructed (based on silicon or other materials) with a positively charged area and a negatively charged area. These two regions are placed close to one another. Photons from the sun hit this material and add enough energy to cause electrons to "jump" from one atom to another, eventually causing them to start flowing from the negative region to the positive region. This flow is harnessed by a flow collector and converted into current.

A more detailed explanation can be found here: http://www.howstuffworks.com/solar-cell.htm