Organic compounds and organic chemistry are thus named for historical reasons, not that they necessarily have anything to do with organisms (although they can). Generally it's carbon-hydrogen based molecules, for example, methane (CH4) is the simplest organic molecule.

https://en.wikipedia.org/wiki/Organic_compounds

As to what is a biological molecule, well, that's a harder question. In a somewhat tautological way I'd probably say that's something coming from a biological organism.