We used the VLA to make deep images of two 7' x 7' fields at 8.44 GHz with 10' resolution. With an rms noise of 3.2 and 5.1 muJy, respectively, in the two fields, we compiled a catalog of 82 sources. From the complete sample of 20 sources with S greater-than-or-equal-to 14.5 muJy, the differential 8.44 GHz source count is dN(S)/dS = (-4.6 +/- 0.7) x S-2.3 +/- 0.2 Jy-1 sr-1 in the range 14.5-1000 mJy. Analysis of statistical image fluctuations from weak sources (Fomalont et al. 1993) suggests that this slope remains unchanged at gamma = 2.3 +/- 0.2 down to approximately 4 muJy. The normalized differential 8.44 GHz counts are similar to those at 1.41 and 4.86 GHz. All show a similarly steep submillijansky slope, which is only somewhat flatter than that expected for a nonevolving Euclidean population (gamma = 2.5). Microjansky radio sources at 4.86 GHz have been identified with faint blue galaxies (18 less than or similar V less than or similar 28 mag). We argue that their expected median redshift is approximately 0.5-0.75. Hence, cosmological evolution may be needed to explain the steep slope of the microjansky counts. The 8.44 GHz counts must converge with slope gamma less-than-or-equal-to 2.0 below S8.44 congruent-to 300 nanojanskys (nJy), or they would exceed the available field galaxy counts down to V approximately 28 mag, and they must permanently converge below S8.44 congruent-to 20 nJy, or their integrated sky brightness would distort the observed thermal cosmic background radiation spectrum at centimeter wavelengths. Radio spectral information is summarized for these sources between 0.33 and 8.44 GHz. The high-frequency spectral indices (S is-proportional-to nu(-alpha)) span the range -2 less than or similar alpha8.4(4.9), alpha8.4(1.4) less than or similar +1.3 with median alpha(med) congruent-to 0.35 +/- 0.15. About 40% of the sources have angular size THETA greater-than-or-equal-to 5'', and the median is THETA(med) congruent-to 2''6 +/- 1''4, or less than or similar 5-40 kpc at the expected median redshift. The extended steep-spectrum sources suggest synchrotron emission in distant galactic disks, while the extended flat-spectrum sources may indicate thermal bremsstrahlung from large-scale star formation, both occasionally with opaque radio cores. The estimated 31.5 GHz sky brightness from nanojansky to jansky levels is less than or similar 36 muK (3 sigma). Even if weak radio sources cluster on scales of degrees as faint galaxies do, their anisotropic contribution to the COBE DMR experiment (with 7-degrees FWHM beam) would not exceed approximately 1.2 muK.
