We study the temperature structure of the intergalactic medium (IGM) using a large cosmological N-body/SPH simulation. We employ a two-temperature model for the thermal evolution of the ionized gas, in which the relaxation process between electrons and ions is explicitly included. In the diffuse, hot IGM, the relaxation time is comparable to the age of the Universe and hence the electron temperature in post-shock regions remains significantly smaller than the ion temperature. We show that, at the present epoch, a large fraction of the warm/hot intergalactic medium (WHIM) has a well-developed two temperature structure, with typical temperature differences of order a factor of a few. Consequently, the fraction of metals in various ionization states such as OVI, OVII, and OVIII, as well as their line emissivities, can differ locally by more than an order of magnitude from those computed with a single temperature model. It is thus necessary to follow the evolution of the electron temperature explicitly to determine absorption and emission by the WHIM. Although equipartition is nearly achieved in the denser intracluster medium (ICM), we find an appreciable systematic deviation between the gas-mass weighted electron temperature and the mean temperature even at half the virial radii of clusters. There is thus a reservoir of warm (Te < 1keV) gas in and around massive clusters. Our results imply that relaxation processes need to be considered in describing and interpreting observational data from existing X-ray telescopes as well as from future missions designed to detect the WHIM, such as the Diffuse Intergalactic Oxygen Surveyor and the Missing Baryon Explorer.