With millions of known organic compounds, classifying them by two independent lenses -- overall carbon-skeleton structure, and functional group -- is what makes the subject navigable at all.
By structure, every compound is first split into acyclic (open-chain/aliphatic, like propane CH3−CH2−CH3, no ring anywhere) or cyclic (containing at least one ring). Cyclic compounds split again by ring composition: homocyclic (the ring is built entirely of carbon) versus heterocyclic (the ring contains at least one non-carbon atom, commonly O, N or S); and each of those splits again by aromaticity. So a homocyclic ring is either alicyclic/carbocyclic (a plain, non-aromatic carbon ring, e.g. cyclopropane) or aromatic -- itself either benzenoid (built around an actual benzene ring, e.g. phenol) or non-benzenoid (aromatic without a benzene ring, e.g. azulene). A heterocyclic ring is either alicyclic (a non-aromatic heteroatom ring, e.g. tetrahydrofuran) or aromatic (e.g. pyridine). Applying the scheme in practice just means asking, in order: is there a ring at all? if so, does it contain a heteroatom? and is it aromatic?
By functional group, compounds are grouped by the reactive atom or group of atoms that gives the molecule its characteristic chemistry -- this is the classification the whole rest of this unit (and every later unit) actually leans on. The book's Table 11.1 catalogues roughly thirty classes: the unsaturated hydrocarbons (alkenes, CnH2n, alkynes, CnH2n−2, defined by their C=C or C≡C bond alone), the halide/oxygen/sulfur single-bonded families (alkyl halides R-X, alcohols R-OH, ethers R-O-R', thiols R-SH, thioethers R-S-R'), the carbonyl-based families sharing a C=O but differing in what else sits on that carbon (aldehydes R-CHO, ketones R-CO-R', carboxylic acids R-COOH, esters RCOOR', acid anhydrides R-CO-O-CO-R', acyl chlorides R-COX, amides R-CO-NH2), the nitrogen- and sulfur-oxide families (sulphonic acids R-SO3H, nitro compounds R-NO2, nitroso compounds R-NO), the amine/nitrile family and its structural cousins (amines R-NH2, nitriles R-CN, and the isomeric group of isocyanides R-NC, cyanates R-OCN, isocyanates R-NCO, thiocyanates R-SCN, isothiocyanates R-NCS), and a handful of nitrogen-nitrogen and other families (imines R-CH=NH, hydrazines R-NH-NH2, hydrazo compounds R-NH-NH-R, phenols C6H5OH, and imides RCON(R')COR'').