Organic chemistry is often thought of as the chemistry of life. In fact, this is a commonly held misconception. Organic chemistry simply means the chemistry of carbon containing compounds, regardless of the origin of those compounds. Whether or not a carbon containing compound comes from life, or even from nature, is irrelevant.
Organic compounds are commonly found in living systems, which is why organic chemistry is so often confused with biochemistry. Proteins, carbohydrates, DNA, fats, and other molecules often associated with the body are all organic. Most drug compounds are organic, and organic chemistry is the heart of pharmaceutical and medicinal chemistry.
The reason carbon plays such an important role in chemistry is that it can concatenate, or form chains. This property makes carbon an excellent building block, a molecular set of tinker toys. In fact, molecular models used by college students and organic chemists resemble childrens' building sets.
Carbon is the star of organic chemistry, but oxygen, nitrogen, and the halogens all play important supporting roles. These atoms make what are called polar compounds. Like a magnet, a polar compound has two opposite poles, although the poles of a magnet are labeled north and south, while the poles of a compound are positive and negative respectively. Polarity influences solubility and reactivity, and chemists who understand the relationship between polarity and structure can often predict the chemical and physical properties of a compound just by looking at the structure. (This saves a lot of time in the lab!)
Compounds containing "heteroatoms" (non carbon atoms other than hydrogen) are more reactive and therefore more interesting than simple hydrocarbons. Hydrocarbons are interesting as fuels and in forming polymers, but otherwise, they must be converted into something else before becoming of use.
The process of changing the structure of a molecule is called organic sysnthesis. An organic chemist will have an arsenal of organic reactions which can be used to change the starting material into the desired product. Coming up with w synthesis is a challenging enough task, but being able to carry it out requires both knowledge and skills that can only be learned by doing.
Organic chemistry is the direct descendant of alchemy, and is still, in many ways, poorly understood. That's why it can be such a trial and error process. The theoretical foundation has not yet been laid to allow truly optimal synthesis to be devised without trial and error. Perhaps someday physical organic chemistry will get to that next step, but for now it remains a long and frustrating road from starting material to product.