We present a comprehensive analysis of the Einstein data base on the Small Magellanic Cloud (SMC). A discrete source search of a 32 deg2 field in the vicinity of the Cloud reveals 70 sources, 22 of which are reported here for the first time. In all, 24 sources have been classified as objects in the SMC, and 13 as Galactic stars, AGNs, and clusters of galaxies. These detected discrete emitters can account for less than half of the excess emission associated with the Cloud. The extensive diffuse emission has an X-ray spectrum substantially softer than that of the discrete sources, consistent with the presence of X-ray-emitting coronal gas associated with the SMC. We have conducted a detailed study of the discrete sources. In particular, we find that an X-ray source varies in its X-ray luminosity by a factor larger than 10 over a period of about 1 yr and has a relatively hard X-ray spectrum which apparently changes with the X-ray luminosity. We suggest that this source, associated with a B1-type star, is a new high-mass X-ray binary in the Small Cloud. Two pointlike X-ray sources are found to have extremely soft spectral characteristics: one is probably a new member of a unique class of low-mass X-ray binaries in the Clouds; and the other is coincident in position with a planetary nebula N67 (SMP 22) in the Small Cloud. The central star of the planetary, most likely responsible for the X-ray emission, has an effective temperature of approximately 3 x 10(5) K and is the hottest ever observed in an X-ray band.