A compressor tank check valve is a one-way or a non-passing valve. In other words, whatever is in the line that the check valve is installed in, can only flow through the check valve in one direction.
Please see the photo below that shows the typical location of a compressor tank check valve. The description as to what they do and why they are important follows the photo.
The red captions in the photo identify the line from the compressor head to the tank, the line from the fitting out to the unloader valve, and the approximate location of the check valve.
As the compressor is running, as air is being compressed, it flows from the pump head down into the tank through the tank check valve. Since the check valve is a one way valve, once compressed air gets into the compressor tank, in theory, the only way it should get back out, is via the discharge coupler into which you connect your air hose.
Upstream from the tank check valve, and typically in the same fitting, is an air line that runs over to the pressure switch and to the unloader valve.
When the compressor stops after reaching cut out pressure, as the pressure switch trips to cut power to the compressor motor, it commonly also toggles the unloader valve to open.
The tank check valves purpose is to keep the compressed air in the tank so it there to be used by downstream air-using equipment.
At the same time, the pressure switch (usually it is a toggle on the pressure switch, but not always) closes the unloader valve. Now the compressor is running, air is being pumped into the air tank, and none of it can flow down the line and out the unloader valve.
When the pressure in the tank again reaches the cut out pressure setting on the pressure switch, the pressure switch trips again, the motor is shut off, the unloader valve is opened, and the cycle repeats.
While a leaking unloader valve is a possibility, the more frequent occurrence is that the check valve has not seated properly, and is allowing air to bleed out of the tank, when normally it should be shut tight allowing no compressed air to escape.
Depending on how much debris was in the check valve seats or the level of check valve failure that had occurred, would determine whether the leak out of the unloader valve was slight or a lot of wasted compressed air.
If you have a leaking unloader valve, first make sure the compressor power is off, that the tank is drained of all compressed air, then remove and clean or replace the check valve assembly to stop the leak.