ISO/LWS grating observations of the ultraluminous infrared galaxy Arp 220 shows absorption in molecular lines of OH, H$_2$O, CH, NH, and NH$_3$, as well as in the [O I] 63 $\mu$m line and emission in the [C II] 158 $\mu$m line. We have modeled the continuum and the emission/absorption of all observed features by means of a non-local radiative transfer code. The continuum from 25 to 1300 $\mu$m is modeled as a warm (106 K) nuclear region that is optically thick in the far-infrared, attenuated by an extended region (size 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 and the extended region. The H$_2$O column density is also high toward the nucleus and lower in the extended region. The column densities in a halo are similar to what are found in the diffuse clouds toward Sgr B2 near the Galactic Center. Most notable are the high column densities found for NH and NH$_3$ toward the nucleus, with values of $\sim1.5\times10^{16}$ cm$^{-2}$ and $\sim3\times10^{16}$ cm$^{-2}$, respectively. A combination of PDRs in the extended region and hot cores with enhanced \hdo photodissociation and a possible shock contribution in the nuclei may explain the relative column densities of OH and \hdo, whereas the nitrogen chemistry may be strongly affected by cosmic ray ionization. The [C II] 158 $\mu$m line is well reproduced by our models and its ``deficit'' relative to the CII/FIR 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.Peer reviewed