A hatchet was a small handheld axe. The word could be used as a verb to describe chopping.
One common idiom was burying the hatchet, meaning to cease arguing. (TOS: "Day of the Dove", TNG: "The Icarus Factor", DS9: "Crossover", "Badda-Bing, Badda-Bang")
Will Riker told a joke that used the similarities between "hatching" and "hatchet": "And just as the headsman swings his axe and the blade is about to fall, the count says, wait, wait, I'll talk! But it was too late. And the moral of the story is? Never hatchet your Counts before they chicken." (TNG: "Data's Day")