We present an analysis of a deep (1$sigma$=13 $mu$Jy) cosmological 1.2-mm continuum map based on ASPECS, the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field. In the 1 arcmin$^2$ covered by ASPECS we detect nine sources at $>3.5sigma$ significance at 1.2-mm. Our ALMA--selected sample has a median redshift of $z=1.6pm0.4$, with only one galaxy detected at z$>$2 within the survey area. This value is significantly lower than that found in millimeter samples selected at a higher flux density cut-off and similar frequencies. Most galaxies have specific star formation rates similar to that of main sequence galaxies at the same epoch, and we find median values of stellar mass and star formation rates of $4.0times10^{10} M_odot$ and $sim40~M_odot$ yr$^{-1}$, respectively. Using the dust emission as a tracer for the ISM mass, we derive depletion times that are typically longer than 300 Myr, and we find molecular gas fractions ranging from $sim$0.1 to 1.0. As noted by previous studies, these values are lower than using CO--based ISM estimates by a factor $sim$2. The 1,mm number counts (corrected for fidelity and completeness) are in agreement with previous studies that were typically restricted to brighter sources. With our individual detections only, we recover $55pm4%$ of the extragalactic background light (EBL) at 1.2 mm measured by the Planck satellite, and we recover $80pm7%$ of this EBL if we include the bright end of the number counts and additional detections from stacking. The stacked contribution is dominated by galaxies at $zsim1-2$, with stellar masses of (1-3)$times$10$^{10}$ M$_odot$. For the first time, we are able to characterize the population of galaxies that dominate the EBL at 1.2 mm.
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