Fossils are items that are dotted around the worlds that you have to find as you explore the worlds.
There are many item 'types', such as Fossils, Minerals, Gems etc.
For fossils, you collect 'bits' of each animal as you explore. For example 'Legs', 'Skulls' etc.
The pirate will generate a list of items that he wants for the ransom, and you can decide once you have found some of those items to either give them towards the ransom, or sell them on the market to get $$ to upgrade and enhance your aliens to make them stronger, faster & better.
If you manage to get at least 1 of each of the bits to complete a whole fossilized animal, then that is worth much more on the market than the individual bits.