Organic chemistry technically just means carbon chemistry.
In biology carbon is the backbone molecule for all life/biological chemistry, although there are theories that on other planets/ecosystems other similar chemicals, such as silicon or germanium (both able to form 4 bonds, like carbon - same column of the periodic table) may be able to fill the same role.
From the scientists perspective discovering 'organic molecules' is important because basically when you get down to the nuts and bolts of what carbon molecules are, the point is that on a planet with an environment such as mars', without 'biology' as an underlying mechanism for the generation of these molecules, there shouldn't be any, I.e. they will all get degraded by the environment without some process churning them out.
That doesn't mean it's the only way that they can be made, just a theory that we have for their source.
The presence of methane (organic and usually biological in origin) in the Martian atmosphere has long been suspected of being persistent (ir should otherwise escape/be broken down) because of renewal due to biological activity however that is far from clear and more likely scenarios put the gas escaping from geological processes under the surface as the cause.
The long and short is thus that the discovery of organic molecules may be a tantalising clue to the presence of biological processes, either now or at some time in the past, however there would likely remain many alternative hypotheses to rule out and it would be (extremely) unlikely that any announcement could point to the existence of a form of 'life' on mars due to the equipment they have available.
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.
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.
Gasoline is an organic molecule. (Quoth wikipedia: An organic compound is any member of a large class of gaseous, liquid, or solid chemical compounds whose molecules contain carbon.)
DNA is a biological molecule - a molecule used more or less exclusively in biology and not spontaneously occurring outside the context of life.
That's all true, but gasoline is actually a biological derivative as it's formed from the remains of living things. So your two examples actually both occur within the context of life.
We do know that organic molecules with non-biological origins do exist on other planets and even on asteroids. Titan has large quantities of organic compounds on it's surface that have non-biological origins, for example. It is believed that at one time Mars had a much thicker atmosphere and active geology which may have lead to the formation of organic compounds.
Right, I was actually thinking of synthetic hydrocarbons (for example motor oil), and thought gasoline was something people could relate to.
Methane on Titan would be a better example, and I actually initially used methane - but most people have experience with methane in the context of biology, too.
