  We study the properties of cosmological shock waves identified in
high-resolution, N-body/hydrodynamic simulations of a $\Lambda$CDM universe and
their role on thermalization of gas and acceleration of nonthermal, cosmic ray
(CR) particles. External shocks form around sheets, filaments and knots of mass
distribution when the gas in void regions accretes onto them. Within those
nonlinear structures, internal shocks are produced by infall of previously
shocked gas to filaments and knots, and during subclump mergers, as well as by
chaotic flow motions. Due to the low temperature of the accreting gas, the Mach
number of external shocks is high, extending up to $M\sim 100$ or higher. In
contrast, internal shocks have mostly low Mach numbers. For all shocks of
$M\ge1.5$ the mean distance between shock surfaces over the entire computed
volume is $\sim4 h^{-1}$ Mpc at present, or $\sim 1 h^{-1}$ Mpc for internal
shocks within nonlinear structures. Identified external shocks are more
extensive, with their surface area $\sim2$ times larger than that of identified
internal shocks at present. However, especially because of higher preshock
densities, but also due to higher shock speeds, internal shocks dissipate more
energy. Hence, the internal shocks are mainly responsible for gas
thermalization as well as CR acceleration. In fact, internal shocks with $2 \la
M \la 4$ contribute $\sim 1/2$ of the total dissipation. Using a nonlinear
diffusive shock acceleration model for CR protons, we estimate the ratio of CR
energy to gas thermal energy dissipated at cosmological shock waves to be
$\sim1/2$ through the history of the universe. Our result supports scenarios in
which the intracluster medium contains energetically significant populations of
CRs.
