1.
deep pool, deep spot in a river
A deep, still section of a river or stream where the water is calm on the surface but deep below. Contrasts with 瀬 (shallow, fast-flowing section).
2.
abyss, depths
Used figuratively for a deep, dark, or unfathomable state, such as the depths of despair or a perilous situation.
Usage
In the literal sense, 淵 refers to a deep, quiet section of a river, as opposed to 瀬 (a shallow, fast-flowing section). The proverb 淵瀬を変える (shallows change to deep pools) describes the unpredictability of fortune.
The figurative sense (brink, verge) is extremely common: ~の淵 (the brink of ~). This pattern is used with negative situations like 絶望の淵 (the brink of despair), 死の淵 (the brink of death), and 破滅の淵 (the brink of ruin).