Fatty acids are long chains of carbon molecules that have hydrogen molecules attached to them (which is why they’re called “hydrocarbon chains”). Fatty acids can be anything from three carbon atoms to about 20 carbon atoms long. The regularity with which hydrogen atoms are attached to the carbon atoms in the chain determines whether the fatty acid is saturated or unsaturated.
A carbon atom, you see, has four bonding sites. When it’s arranged in a chain, two of those bonds are automatically taken up, one to either side. That leaves two bonds, which can either be used to bond to hydrogen atoms, or to double-bond to the carbon atom next door.
When the carbon atoms in a fatty acid chain have all their available bonds taken up by hydrogen atoms, the fatty acid is said to be “saturated”. Anything less than that and the fatty acid is said to be “unsaturated” (if the entire chain is missing only one hydrogen atom, the fatty acid is “monounsaturated”, anything more than that is “polyunsaturated”). A handy visual aid can be found here.
It’s the level of saturation that determines a fat’s physical properties (hardness, melting point, etc.). Saturated fatty acids, as you can see from the picture, are pretty orderly, pretty straight. Attached to a glycerine backbone, they form straight and orderly molecules that stack on top of each other pretty well. And in the molecular world, good stacking ability means they form crystals easily, and lots of crystals means the fat is solid at room temperature.
An unsaturated fat is a different story. Here, one or more hydrogen atoms are missing from the chain. In that case, pairs of carbon atoms (and it always happens in pairs) are double bonded to each other (again, look at the visual aid). It’s a perfectly valid arrangement, yet it does put a kink in the chain. Attached to a glycerine backbone, these unsaturated fatty acids make molecules that don’t stack nearly as well as saturated fat molecules. Bad stacking ability means very few crystals, which means the fat is liquid at room temperature.
As you can pretty well guess by this point, animal fats are by and large the saturated ones, plant fats are the unsaturated ones.