'Organic' is not the correct term for what is generally marketed as 'Organic Food'. In physical terms, Organic is simply something that has a compound of carbon as its primary building block. In market terms, Organic food refers to 'natural' food. This is food that is as free of modern artificial chemical-based additives as is possible.

How Organic Food Is Certified:

Each country has a different specification of what qualifies as organic food. In the US, for example, the food must be 95% natural. As is clear, this is a rather difficult test for what is "natural". As such, a more clear definition is presented by looking at the source of the food.
The food must be grown (in case of vegetables, fruits and crops) without any chemical based pesticides, insecticides or herbicides. It must also be free of any and all chemical fertilizers, human waste and sewage sludge.
There are additional conditions on the processing of this food. When the food is being processed, cleaned and refined, it must be exposed to ionizing and radiation treatments, nor must it be embellished with any food additives.
All of these conditions ensure that only what is completely natural goes into the consumer markets.

A different set of regulations apply to the meat and animal rearing industries. The animals being used (in dairy, for eggs, or meat) must be free of artificial growth hormone additions, antibiotics, and any genetic modifications. Additionally, all the food that they are brought up on must also be organic in nature.

The genetic controversy::

There was a major debate about whether genetically modified organisms should be classified as natural or not, since (as the argument stands), nothing artificial is being added to the animal's system.

This flawed reasoning stems from a lack of understanding about what genetic modifications are intended for. Genetic modifications induce changes in the very genomic make-up of the animal that wouldn't otherwise exist. This requires over-expression of certain traits (for example, to produce more milk), or even complete gene splicing. All of this leads to changes in the genome which would otherwise never be naturally fixed in the population. In the end, it was suitably proven that genetically modified organisms are just as "artificial" as chemically induced animals.

Due to such stern conditions, it is illegal to sell organic food (or market it as such) in many countries without proper certification.

The biggest two challenges faced by organic food production are:
1. The yield is not large enough to sustain the entire population in the world.This is due to lack of sufficient nitrogen in the soil, increased transportation costs on a small farm and so on.
2. Inability to know what food is truly organic.

The Yield:
Many researchers and scientists argue that the world could at most produce only enough food organically to feed 4-4.5 billion people. On the other hand, to even get this yield, mass deforestation would have to take place, since the cattle required to maintain a nitrogen rich soil would take over most of the forested area today.
As such, many question the viability of organic food.

At the same time, it is true that organic food production uses less energy overall than its counterpart. This is because it maintains more ecosystems on the farm. However, part of this is because organic food production is currently limited to small farms where the buyers make individual purchases.
As it becomes more commercialized, this may not hold true. Essentially, there are many variables relating to Organic food production that simply cannot be tested.

The Trust:
This is a more important issue at the consumption level. How can one know what is really organic and what is not? In the past, buyers would develop a relationship with the various farmers, and personally observe the farming methods used to ensure that the food is indeed organic. However, as the demand for organic food rises and greater organic food delivery is desired, this is not a feasible system.
As such, many nations have third-party testers who certify various food production as organic. Since it is nearly impossible for most people in cities such as Paris, Delhi, Tokyo, New York and Sydney to personally observe the food production, various seals are given denoting how organic a food product is.
For example, this is Australia's seal:


And this is the international seal:


Conclusions:

Many studies have shown conclusive evidence that organic food is better for the environment and doesn't contain many of the unnecessary and harmful chemicals. Not many studies have been able to study the effects of widespread organic farming and effects on health, so much of that is still based in speculation and experimental models.