A warp core was the energy production reactor technology which allowed starships to use warp drive to travel at warp speeds.
Commonly, warp cores utilized matter/antimatter reactions to produce the required energy to achieve warp speed. Federation starships used deuterium and antideuterium which was reacted in a dilithium matrix. Romulan starships typically used artificial quantum singularities to produce the required power. The majority of starships used their warp cores for general energy requirements in addition to powering their warp drives.
An unusual hybrid starship encountered by the USS Enterprise-D in 2370 included a warp core from a Romulan warbird. (TNG comic: "Strategy")