Images not appearing correctly.

I am making a program and I have found good images to use in it. However, My program will only allow 16-color images. So what I tried to do is download the 256 color animated GIF’s (of sonic, tails and knuckles) and then I tried to convert them into16 colors but the colors dont seem to turn out correctly. Is there a small application < 1MB that can do this well where I can still see the original colors? For example, I tried to convert a picture of DR.Robotnik to 16 colors. In the 256 color mode, I saw white. When I changed it to 16 color mode, I see gray. I should still see white. No, It isnt my monitor. No, it isnt a virus. What can I do to make images appear correctly in both modes?