What is the function of the first stage connector regulator?

The first part of the regulator is the first stage connector hose.  The attached valve helps lower air pressure from the scuba tank so instead of 2,000 psi gushing out, flow would be significantly lowered and controlled.  Pressure chambers exist inside the first stage regulator that change the airflow rate to ensure that devices using your air function appropriately for the depth of the dive.  As you use the mouthpiece to inhale air, pressure within the hose is lower than ambient water pressure.  This causes pressure to be pushed into a piston or diaphragm that attaches to the valve so airflow is opened up.  With that, the air you breathe while diving is balanced at all times.