Several small emerging bipolar regions have been observed with the Advanced Stokes Polarimeter (ASP), including extensive time series measurements of one small region. Both new and previously recognized properties of the actual site of first emergence, where the magnetic field is nearly horizontal to the surface, are revealed by these observations. They provide the most complete and accurate observational description to date of newly emerging vector magnetic fields. We find that: 1) the strength of the magnetic field at the site of the emergence (where the vector field is nearly parallel to the solar surface) ranges from about 200 to 600 G, 2) as individual flux elements migrate rapidly away from the emergence zone, they attain kiloGauss strengths only after becoming oriented nearly vertically, 3) the emergence zone is dotted by small, transient, upward rising ( ~ 1 km s(-1) ) horizontal magnetic elements as indicated by the Doppler shift of the polarized spectral profiles, 4) the leading polarity flux coalesces immediately into a compact region which forms a pore, but the emerging following polarity flux is spatially much less compact, 5) some ``moving magnetic features'' having the same magnetic polarity as the growing pore, but on the opposite side of the pore from the emergence zone, coalesce with the pore during the observation period, and 6) the observations suggest a low canopy of weak horizontal magnetic fields arches over the emergence zone. These observations support a widely accepted picture of emerging bipolar flux: the buoyantly rising flux transports mass from the photosphere into the chromosphere, where it then may drain downward along arched magnetic loops. The observed formation of a pore suggests that emergence of subsurface structure, not organized flows near the surface, is largely responsible for the apparent coalescence of sunspots from more diffuse fields viewed at the solar surface. These observations neither confirm nor refute the operation of convective collapse of flux tubes.