Mudstone = The common rock.
Slate = A stone that is made of layers. Often easy to break little by little.
Limestone = The vary in species makes limestone hard to find.
See the wiki page for info how to tell it apart from mudstone and granite.Marble = N/A
Granite = A rather mixed up looking stone. Easy enough to find, hard to obtain.
Claystone = A rock composed mostly of clay. Balanced attitude.
Bassalt = A rare rock with a fiery attitude. Not a good first pet.
Sandstone = Generally a sandy chunk of rock. Sandstone may be observed in the Grand Canyon. A rather quiet stone that prefers dry living spaces with a single pool of water.
Fossils = Bones and other objects get stuff in stone for a long time, creating a fossil. Depending on the fossil, your pet may have different living preferences. Non animal fossils are considerably flexible, although preferring moist dirt as living space. Fish fossils and such prefer liquidy habitats, while others prefer thick mud. Bird fossils like a mix of mud and grass. Be sure to put some twigs with your bird fossil for high perches. Each fossil will act like the creature, plant, or other thing that it, well, is. Be sure to get a rough estimate of what your fossil is. In the case of not knowing, give it mud and some sort of dry land. This should keep most fossils content, though not super-happy. So long as you play with your fossil, it wont care.
Gems = Quartz, Crystals, and any precious stone is in this category. Often bought from shops, these are often used to human hands. Wild gems can be unpredictable.