Grease was an oily substance used as a lubricant. It could be made of a variety of materials.
In 2370, when Benjamin Sisko asked Miles O'Brien to allow Jake Sisko to shadow him, O'Brien assured Sisko that he would keep Jake "up to his elbows in thorium grease." (DS9: "Paradise")
In 2372, Neelix discovered the Vulcan holiday Rumarie, which involved men and women covered in Rillan grease. (VOY: "Meld")
In 2373, Leonardo da Vinci was surprised that Captain Kathryn Janeway was willing to help him repair a machine even though her hands would get covered in goose grease. (VOY: "Scorpion")
Lieutenant Tom Paris created a holoprogram called Grease Monkey, a reference to the slang term associated with car mechanics due to the grease that covered their hands after working on engines. (VOY: "Vis à Vis")