  We discuss the origin and physical nature of the Lya forest absorption
systems as found in hydrodynamical simulations of the Intergalactic Medium in a
standard Cold Dark Matter cosmology (Omega=1, H0=50 km/s/Mpc, sigma_8=0.7). The
structures of the systems that give rise to the Lya forest span a wide range in
morphologies, depending on the density contrast. The absorption properties of
the forest clouds correlate strongly with those of the underlying physical
systems from which they arise. The highest column density systems (log N_HI >
15), correspond to highly overdense spheroidal structures, moderate column
density systems (13 < log N_HI < 14), correspond to filaments, and the lowest
density absorption systems originate from discrete fluctuations within
underdense regions a few megaparsecs across, cosmic minivoids. Most of the
intergalactic HeII opacity arises from these underdense regions. We are able to
account for the distribution of optical depths in our synthesized spectra
entirely by absorption due to discrete systems. We find that virtually all the
baryons in the simulation fragment into structures that we can identify with
discrete absorption lines, with at most 5% remaining in a smoothly distributed
component (the Gunn-Peterson effect). We compare our results with the cloud
ionization parameters inferred from Keck HIRES measurements of carbon and
silicon in the Lya forest. Combining with constraints imposed by measurements
of the mean intergalactic HI opacity, we find 0.03 < Omega_b < 0.08 and 0.3 <
(Gamma_HI/ 10^-12 s^-1) < 1 at z=3-3.5.
