The brink of something is the very edge of it. If you want to give your mom a good scare, stand on the brink of a steep cliff and slowly lean forward.
The noun brink describes the edge of a steep drop-off or slope, or the edge of a boundary marking where something begins, like the brink of a pond. Brink also describes the figurative edge or start of something. Nations on the brink of war are very close to mobilizing troops; if you are on the brink of losing your temper, you might try walking away or slowly counting to ten.