A hatchet is a little ax used for chopping wood or for hewing a round log into a squared-off shape. If you go camping, you might take a hatchet with you.
If someone hands you a hatchet and tells you to start splitting a pile of firewood, they expect you to chop each log into smaller pieces. The difference between a hatchet and an ax is that hatchets have short handles and are meant to be held in one hand. Axes have longer handles for two-handed chopping. If someone wants to "bury the hatchet," it's not a threat—they just want to end an argument and be friends.