Fossils are the really, really old remains of a plant or animal — so old they've turned to stone. Fossil is also an insult for an old or old-fashioned person.
Fossils are very important to scientists who study animals: they're one of the ways we learn about the beasts and critters of the past. A fossil is an imprint of the bones of that animal in rock or stone. If you see a reconstruction of a dinosaur in a museum, scientists used fossils to figure out how it should look. Also, calling someone a fossil is like calling them a geezer or old fogey. It's not a compliment.