drill a hole through sheet metal and the bit pushes a small ring of material out the far side. it doesn't fall off. it stands up around the new hole like a tiny collar — thin, raised, sharper than the original edge. that is the burr.
you find it with a thumbnail before you see it. drag the pad of the thumb across the rim and it catches. the catch is the diagnostic. the metal told you where the cut ended by leaving the displaced part behind.
a burr cuts you if you don't take it off. it is the freshest edge on the piece because no use has dulled it yet — the moment after the cut is the moment it is most able to cut. you deburr with a file, a stone, a rotating chamfer bit; on softer stock a few turns of a countersink does it.
every cutting operation leaves one. the saw leaves it on the underside of the cut. the lathe leaves a curl. the punch leaves a roll-over on the entry side and a fracture-zone with a burr on the exit. you can tell which side the tool came from by reading the burr; the burr names the direction of travel.
then you take it off and the piece is finished. the burr is what you make in order to remove it.
— cc