We present Herschel Spectral and Photometric Imaging Receiver Fourier Transform Spectrometer (Herschel SPIRE-FTS) observations of Arp 220, a nearby ultra-luminous infrared galaxy. The FTS provides continuous spectral coverage from 190 to 670 mu m, a wavelength region that is either very difficult to observe or completely inaccessible from the ground. The spectrum provides a good measurement of the continuum and detection of several molecular and atomic species. We detect luminous CO (J = 4-3 to 13-12) and water rotational transitions with comparable total luminosity similar to 2 x 10(8) L(circle dot); very high-J transitions of HCN (J = 12-11 to 17-16) in absorption; strong absorption features of rare species such as OH(+), H(2)O(+), and HF; and atomic lines of [C I] and [N II]. The modeling of the continuum shows that the dust is warm, with T = 66 K, and has an unusually large optical depth, with tau(dust) similar to 5 at 100 mu m. The total far-infrared luminosity of Arp 220 is L(FIR) similar to 2 x 10(12) L(circle dot). Non-LTE modeling of the extinction corrected CO rotational transitions shows that the spectral line energy distribution of CO is fit well by two temperature components: cold molecular gas at T similar to 50 K and warm molecular gas at T similar to 1350(-100)(+280) K (the inferred temperatures are much lower if CO line fluxes are not corrected for dust extinction). These two components are not in pressure equilibrium. The mass of the warm gas is 10% of the cold gas, but it dominates the CO luminosity. The ratio of total CO luminosity to the total FIR luminosity is L(CO)/L(FIR) similar to 10(-4) (the most luminous lines, such as J = 6-5, have L(CO,J=6-5)/L(FIR) similar to 10(-5)). The temperature of the warm gas is in excellent agreement with the observations of H(2) rotational lines. At 1350 K, H(2) dominates the cooling (similar to 20 L(circle dot) M(circle dot)(-1)) in the interstellar medium compared to CO (similar to 0.4 L(circle dot) M(circle dot)(-1)). We have ruled out photodissociation regions, X-ray-dominated regions, and cosmic rays as likely sources of excitation of this warm molecular gas, and found that only a non-ionizing source can heat this gas; the mechanical energy from supernovae and stellar winds is able to satisfy the large energy budget of similar to 20 L(circle dot) M(circle dot)(-1). Analysis of the very high-J lines of HCN strongly indicates that they are solely populated by infrared pumping of photons at 14 mu m. This mechanism requires an intense radiation field with T > 350 K. We detect a massive molecular outflow in Arp 220 from the analysis of strong P Cygni line profiles observed in OH(+), H(2)O(+), and H(2)O. The outflow has a mass greater than or similar to 10(7) M(circle dot) and is bound to the nuclei with velocity less than or similar to 250 km s(-1). The large column densities observed for these molecular ions strongly favor the existence of an X-ray luminous AGN (10(44) erg s(-1)) in Arp 220.