An ultraluminous infrared galaxy (ULIRG) is a galaxy emitting more than 1012 times solar luminosity in the far infrared. It can be considered a type of luminous infrared galaxy, the latter defined as having 1011 times solar luminosity in far infrared. A hyperluminous infrared galaxy is a galaxy with 1013 times solar luminosity in far infrared. (Sources also cite these thresholds as applying to the galaxy's full bolometric luminosity, sometimes stipulating that most of its emission is within the infrared or far infrared; I presume this is the typical case for such bright galaxies.)
A LIRG is often a starburst galaxy and/or has an active galactic nucleus or a quasar. The infrared would be from thermal emission at a much lower temperature than stars themselves, i.e., of clouds heated by the young stars they are creating, and if the clouds are dusty, the infrared is the only sign of star formation. A ULIRG is typically a nearby galaxy (low redshift), and often a merger. When distant, it is a useful sign of star formation.