We report on Chandra observations of 18 hard X-ray (>20 keV) sources discovered with the INTEGRAL satellite near the Galactic plane. For 14 of the INTEGRAL sources, we have uncovered one or two potential Chandra counterparts per source. These provide soft X-ray (0.3-10 keV) spectra and subarcsecond localizations, which we use to identify counterparts at other wavelengths, providing information about the nature of each source. Despite the fact that all of the sources are within 5 degrees of the plane, four of the IGR sources are active galactic nuclei (AGNs; IGR J01545+6437, IGR J15391-5307, IGR J15415-5029, and IGR J21565+5948) and four others are likely AGNs (IGR J03103+5706, IGR J09189-4418, IGR J16413-4046, and IGR J16560-4958) based on each of them having a strong IR excess and/or extended optical or near-IR emission. We compare the X-ray and near-IR fluxes of this group of sources to those of AGNs selected by their 2-10 keV emission in previous studies and find that these IGR AGNs are in the range of typical values. There is evidence in favor of four of the sources being Galactic (IGR J12489-6243, IGR J15293-5609, IGR J16173-5023, and IGR J16206-5253), but only IGR J15293-5609 is confirmed as a Galactic source as it has a unique Chandra counterpart and a parallax measurement from previous optical observations that puts its distance at 1.56 +/- 0.12 kpc. The 0.3-10 keV luminosity for this source is (1.4(-0.4)(+1.0)) x 10(32) erg s(-1), and its optical/IR spectral energy distribution is well described by a blackbody with a temperature of 4200-7000 K and a radius of 12.0-16.4 R-circle dot. These values suggest that IGR J15293-5609 is a symbiotic binary with an early K-type giant and a white dwarf accretor. We also obtained likely Chandra identifications for IGR J13402-6428 and IGR J15368-5102, but follow-up observations are required to constrain their source types.