Integral field optical spectroscopy using the INTEGRAL system has been used to characterize the kinematic and ionization properties of the warm gas within 2 kpc of the dust-enshrouded nucleus of Arp 220. Owing to the large internal extinction toward the nuclei, the brightest stellar and line-emitting regions observed at optical wavelengths do not coincide with the dust-enshrouded near-infrared and radio nuclei of Arp 220 but are located northwest of the nucleus at a distance of about 750 pc. Moreover, although the continuum and the line-emitting gas share similar distributions, their emission peaks are displaced, with the Hα emission peak located at about 300 pc southwest of the optical stellar continuum emission peak. A line decomposition analysis has been performed in the complex and high spatially variable emission-line profiles. Three different kinematically distinct and extended gaseous components have been identified in the ionized gas. One narrow component (R) indicates rotation, while the other two components (O and B) are well interpreted by the presence of a biconical outflow. Specifically, the rotational component R traces quiescent gas located in a nuclear disk with the spin axis along the southeast-northwest direction (P.A. 135°). This component of ionized gas seems to be coupled with the 100 kpc disklike H I gas and the 1 kpc molecular disk detected in CO (Scoville et al.). The inclination-corrected rotational velocities imply a dynamical mass (Mdyn) of 2×1010 Msolar within a radius of 1.5 kpc. This relatively high value indicates a large mass concentration in the nuclear region of Arp 220, as Scoville et al. already inferred by the presence of 5×109 Msolar of molecular gas in a nuclear, 0.5 kpc disk. One of the outflow components, O, has peak-to-peak velocities of 1000 km s-1. The broad component B, with an average width of 815 km s-1, is detected at about 600 pc northwest of the dust-enshrouded nucleus and is blueshifted by 300 km s-1 with respect to the system velocity. The two-dimensional distribution and kinematics of the components are consistent with a bipolar cone geometry with an opening angle of about 90° and are perpendicular to the nuclear disk of gas, as expected in the starburst-driven galactic wind scenario proposed by Heckman, Armus, & Miley. In most of the observed regions, the ionization status of the different gas components is consistent with a shock-heated LINER-like or Seyfert 2 nebula as judged by the [N II]/Hα and [S II]/Hα emission-line ratios. Although the [O III] and Hβ lines are undetected in most of the regions, the brightest zone located northwest of the nucleus could be classified as a Seyfert 2 nebula based on the additional constraint given by the measured lower limit (>=5) for the [O III]/Hβ ratio. There is no evidence of excitation gradients along the symmetry axis of the outflow nor of a biconical ionization structure, which suggests that the ionizing field is homogeneous and less collimated than the gas outflow. However, there are four clearly identified extranuclear regions where the [N II]/Hα ratio decreases by a factor of 5 and is close to the typical values of H II regions. None of these regions are in spatial agreement with the star clusters found in the infrared by Scoville et al., confirming that they must be relatively old globular clusters.