Infrared Space Observatory Long Wavelength Spectrometer grating observations of the ultraluminous infrared galaxy Arp 220 shows absorption in molecular lines of OH, H₂O, CH, NH, and NH₃, as well as in the [O I] 63 μm line and emission in the [C II] 158 μm line. We have modeled the continuum and the emission/absorption of all observed features by means of a nonlocal radiative transfer code. The continuum from 25 to 1300 μm is modeled as a warm (106 K) nuclear region that is optically thick in the far-infrared, attenuated by an extended region (2'') that is heated mainly through absorption of nuclear infrared radiation. The molecular absorption in the nuclear region is characterized by high excitation due to the high-infrared radiation density. The OH column densities are high toward the nucleus (2-6 × 10¹⁷ cm^-2) and the extended region (~2 × 10¹⁷ cm^-2). The H₂O column density is also high toward the nucleus (2-10 × 10¹⁷ cm^-2) and lower in the extended region. The column densities in a halo that accounts for the absorption in the lowest lying lines are similar to what are found in the diffuse clouds toward the star-forming regions in the Sgr B2 molecular cloud complex near the Galactic center. Most notable are the high column densities found for NH and NH₃ toward the nucleus, with values of ~1.5 × 10¹⁶ cm^-2 and ~3 × 10¹⁶ cm^-2, respectively, whereas the NH₂ column density is lower than ~2 × 10¹⁵ cm^-2. A combination of photodissociation regions (PDRs) in the extended region and hot cores with enhanced H₂O photodissociation and a possible shock contribution in the nuclei may explain the relative column densities of OH and H₂O, whereas the nitrogen chemistry may be strongly affected by cosmic-ray ionization. The [C II] 158 μm line is well reproduced by our models and its "deficit" relative to the C II/far-IR ratio in normal and starburst galaxies is suggested to be mainly a consequence of the dominant non-PDR component of far-infrared radiation, although our models alone cannot rule out extinction effects in the nuclei.