High-resolution H I observations taken at the VLA of the interacting pair of galaxies NGC 2207 and IC 2163 are presented, compared with optical and radio continuum images, and analyzed with detailed computer simulations (in Paper II) in order to determine the orbits and study the effects of in-plane and out-of-plane tidal forcing during a recent, close, nonmerging encounter between galaxies of comparable mass. IC 2163 has an ocular shape (an eye-shaped central oval with a sharp apex at each end) and a double- parallel arm structure on the side opposite NGC 2207. Our observations of IC 2163 find (1) an intrinsically oval shape to the disk, (2) large streaming motions along the oval, (3) H I tidal arms located symmetrically on opposite sides of the nucleus (even though one side is obscured by NGC 2207 at optical wavelengths), and (4) a line-of-sight velocity difference of 70-100 km s-1 between the two components of the double-parallel arm. Optical surface photometry of IC 2163 indicates that most of the stars in the interarm region have been cleared away and put into the central oval and tidal arms. The kinematic and structural anomalies of IC 2163 are consistent with the predictions of N-body galaxy encounter simulations if NGC 2207 moved approximately in the plane of IC 2163 in a prograde sense. The companion, NGC 2207, shows different types of disturbances. The main body of H I gas in NGC 2207 forms a broad, clumpy ring that contains relatively thin stellar arms and corresponds to a plateau in the radial distribution of optical light. The most massive H I clouds in the ring do not always coincide with the stellar arms. The ring is broken in the south, and a filamentary pool of H I extends 40 kpc farther south. The velocity field in the main disk of NGC 2207 is highly distorted with isovelocity contours that are shaped like an open trailing spiral, and the H I line profiles in this region are very broad with Gaussian dispersions of 40-50 km s-1. The kinematic disturbances in NGC 2207 suggest that the main tidal force on NGC 2207 was perpendicular to its disk. The companion side of the H I ring is unusually bright in λ20 cm radio continuum emission, perhaps indicating a shock front. There is no significant excess of star formation in these galaxies, but there are several 108 Msun gas clouds in each disk, some of which may eventually become detached dwarf galaxies.