Organic chemistry is the chemistry of carbon compounds. Carbon is special because it is tetravalent and shows catenation — its atoms join one another in long chains, branches and rings. A carbon atom with four single bonds is $sp^3$ hybridised and tetrahedral (bond angle $109.5^\circ$); with one double bond it is $sp^2$ and planar ($120^\circ$); with a triple bond it is $sp$ and linear ($180^\circ$).
Structural representations
The same molecule can be drawn three ways. The Lewis (complete) structure shows every atom and bond; the condensed formula omits some bonds (e.g. $CH_3CH_2CH_2OH$); the bond-line (skeletal) formula draws only the carbon skeleton as a zig-zag — each line-end and vertex is a carbon, and hydrogens on carbon are understood. Bond-line drawings are quick and are the working language of organic chemistry.
Classification
Compounds are first split into acyclic (open-chain) and cyclic (closed-chain). Cyclic ones are homocyclic (only carbon in the ring, e.g. benzene) or heterocyclic (a hetero-atom such as N or O in the ring, e.g. pyridine). A functional group is the reactive atom or group that decides chemical behaviour — $−OH$ (alcohol), $−CHO$ (aldehyde), $−COOH$ (carboxylic acid), $−NH_2$ (amine), and so on. A homologous series is a family with the same functional group and a constant difference of $CH_2$ (14 u) between successive members; they share a general formula and graded physical properties.
IUPAC nomenclature
A name has three parts: root word (chain length: meth-, eth-, prop-, but-, pent-…), a suffix for the principal functional group (-ane, -ene, -ol, -al, -one, -oic acid) and a prefix for substituents (methyl-, chloro-, nitro-). Steps: pick the longest chain containing the functional group; number it so the principal group gets the lowest locant; name and number substituents alphabetically; combine. Thus $CH_3CH(CH_3)CH_2OH$ is 2-methylpropan-1-ol.
Isomerism
Isomers have the same molecular formula but different structures. Structural (constitutional) isomers differ in connectivity: chain, position, functional-group and metamerism. Stereoisomers have the same connectivity but different 3-D arrangement — geometrical (cis–trans) from restricted rotation about a double bond, and optical from a chiral carbon (four different groups), giving non-superimposable mirror images.