Fats are deceivingly simple molecules. fats are just atoms of carbon linked together in a chain.
Assuming hothing is attached to either end of the chain (a free fatty acid) you will find a carbon surrounded by hydrogen (CH3) on one end, and on the other end you will find a few oxygen (COOH or COO-).
Surrounding all the carbons are hydrogen atoms. Now what gives various fats most of their biological character is the length of the chain and the number of double bonds.
The more carbons the longer the chain. A double bond is what you get when you take away a few hydrogen and the bond "doubles up" on the carbon. These double bonds are very important and dictate what type of fat it is and effect it will have on the body.