A hyperluminous infrared galaxy (HLIRG or HyLIRG) is a galaxy emitting more than 1013 times solar luminosity in the far infrared. A HLIRG can be considered a type of luminous infrared galaxy (LIRG), i.e., a galaxy with 1011 times solar luminosity in far infrared. A ultraluminous infrared galaxy has 1012 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.