What is the purpose of the bypass hose?

I will assume that is in an automobile. A bypass hose goes around the thermostat in your cooling system to allow some coolant to flow whenever the engine is running.

This prevents to much pressure to be built up in the system, allows for even cooling at start up and even gives you a slight amount of heat when you start the engine. Your heater core is often part of the bypass system in a car.

When the engine's thermometer opens the flow of coolant in the car increases this cooling the engine. As the engine cools the thermostat closes slightly and increases the flow through the core and bypass hose.