Refers to a small break, cut, or flaw on a surface, such as on a dish, blade, or edge. Often used in compound forms like こぼれ目 (a crack or flaw).
This teacup has a small chip.
The blade got a nick.
Derived from the verb こぼれる (to spill, to chip). The noun form こぼれ refers to the result of chipping or spilling over, hence a small break or flaw.