The depth limit, also referred to as max depth, is the measure of how deep below the water surface your shark can swim before being subjected to accelerated health drain caused by the water depth. Depth limits differ from shark to shark, and don't apply to some sharks at all. For example, the Great White Shark, Megalodon and Big Daddy all lack depth limits.
The depth limit is merely an indicator; it does not directly deal damage. However, the mechanic linked to the depth limit, which is pressure damage, does. Pressure damage amplifies the rate at which your shark's health drains naturally, for as long as the shark is below the maximum depth which it can withstand. Getting hurt by going past the depth limit gives you the accolade "Too deep! Pressure damage"
The depth limit has no effect on Top-Secret Lab sharks, the Great White Shark or greater from Hungry Shark Evolution.