Having a hard time finding out what this stone is called.
The surface has been polished which revealed these "layers" with some degree of translucency.
Your rock is most likely agate. Try to scratch the polished surface with a nail, and then wipe the mark. Do you still see a scratch? If there is no scratch, the the rock is made of silica therefore it is agate. Also search "agate" here on this website for some more information and identification techniques.