Definition of "organic" in Science
In science (esp. in chemistry), organic matter loosely refers to matter that eventually make up living beings are are remains of living beings. Typically organic matter consists of long chains of carbon surrounded by hydrogen atoms and a few oxygens and nitrogens (and eventually other things such as sulfur and phosophorus).
Can we eat something that is not organic?
Unlike plants, we (all animals and fungi) are heterotrophs (feed on organic matter only). As a consequence, there is no such thing one could call food that would not be organic.
The (mis)use of the term "organic" in popular culture
In the popular culture organic is a term that tries to refer to certain type of food that are grown under certain conditions and treated a certain way. However, there is really no good definition of organic in the popular culture and there exists no objective definition that would satisfy someone that wishes to categorize food into what they like to call "organic food" and "not organic food". Similarly there is no good objective definition of GMO either.
Some organization may make up definitions of "GMO" or "organic food" but those definitions rely on a long list of products rather than to an actual definition.
I just want to attract the attention of the naive reader that I am not saying that the term organic as used in the popular culture is healthy/not healthy or good/bad. Arguments concerning whether it is ethical to buy organic food goes both ways and it is not the job of a science website to discuss questions of ethics.
Are GMO safe to eat?
You are not asking this question here. But in case you wonder you might want to have a look at the Skeptics.SE post: Is Genetically Modified food safe to consume?