What do organic compounds contain that inorganic compounds do not?
What seperates the two is the fact that compounds containing carbon atoms are classified as organic, and those without carbon are inorganic.
Carbon is the backbone for organic compounds as it can maintain 4 covalent bonds. All living things, that we know of, make use of this characteristic. Therefore, you will often describe living things such as animals, plants, fungi, bacteria, and protozoa as organic.
Fun fact: If you look at a periodic table you'll see that silicon will also make 4 covalent bonds, so why are we made of carbon and not silicon? Because relatively, it is too heavy.