What types of light can't a plant photosynthesize in?
Chlorophyll can use a quite broad range of light for photosynthesis, the only range where it is not working is approximately between 500 and 620nm. See this adsorption spectrum of chlorophyll (from the Wikipedia article on Chlorophyll):



The lack of chlorophyll to absorb light between 500 and 620nm (roughly) results in the green color of leafs, because this light is reflected.
Besides the wavelength also the energy of the light is important, so plants will grow better in sunlight than in the light of your monitor. This link about the maximum effectiveness of photosynthesis is also interesting.