Organic chemistry is the study of carbon chemistry. The term "Organic" is a hold over from way back in the 1800's when it was believed that the molecules of life could only come from life -- a sort of distinction between the matter of inanimate objects and the matter of living beings. This isn't too far off: In fact, Carbon behaves in ways that no other element does. So much different, in fact, that it and water are considered necessities when searching for life. The main reason Carbon is thought necessary for life is that it is the only element that can form exceptionally long chains -- up to millions of atoms in a single molecule! The molecule closest to it in chemical nature is Silicon, and the longest chain of silicon is roughly around 10 Silicon atoms long. However, the distinction is untrue. It was disproven when a chemist by the name of Wohler (that "o" is actually an omlaut) created a compound known as Urea from ammonium and cyanic acid. Urea was a well known compound in Wohler's time that originated from mammalian urine (hence urea), but both ammonium and cyanic acid were classified as inorganic compounds, essentially showing that organic matter can originate from inorganic.
When studying organic chemistry, there is an analogy I like to employ. That analogy is Legos. I played with Legos as a kid quite a bit. And it's funny, but Legos, made of Carbon, create a great analogy for Carbon and all of the elements involved in the chemistry of Carbon. When studying organic chemistry, we classify different combinations of elements into "functional groups". When certain elements are bonded to other elements in a certain pattern, they exhibit similar traits -- observable chemical traits. They interact with each other in certain ways, break off, form new bonds, and become new compounds. Essentially, each group is like a Lego: You have small, stout blocks that build the basis for many Lego structures -- the Carbon atoms of Legoland -- and you have long blocks, similar to long "R" carbon chains ("R" just denotes "Carbon chain"). You have specialized blocks that can only fit in certain places, like the cannons, or the flags and flag poles, or the little switches. These are similar to the other functional groups in organic chemistry: They all behave in a certain way (due to their chemical make up) and can only attach to other groups because of their behavior (Think of the castle gates: Feasibly, they can serve as gates, or grates on the ground, but they aren't very good rockets for your space Lego sets).
Carbon chemistry, in this sense, is just playing with Legos. Except, with chemistry, you can't use your hands to pick a block off and re-stick it somewhere. The building blocks are too small. You have to figure out ways to interact with the building blocks without picking them up and putting where you want them: And this is where reaction mechanisms come in. Mechanisms, as a whole, are a step-by-step diagram of what occurs on the chemical level during the process of a chemical reaction. You show where electrons move from and to, what charges various elements have (which can attract or repel electrons), and which elements attach to other elements. The electrons in carbon chemistry can be compared to the nubs on top of the blocks in Legoland -- they keep the blocks together. Armed with the knowledge of a given element or functional group's tendencies, you can pick apart carbon groups, reattach other functional groups, and end up with something entirely different. How that different thing behaves and how you get there depend on a lot of things -- more than I want to go into with this particular blog post -- but the basic analogy holds. You pick apart the blocks of life, and reattach those blocks to build a shape that, in Lego talk, may have started as a house, but is now an attack helicoptor.