This silicone check valve, commonly called a Thomas valve, is a small one-way valve used in both party pumps and standard Sankey-style keg couplers. In a party pump, pressing the handle forces air through the valve and into the keg, and that air pressure is what pushes beer up through the tap.
When the handle springs back or pumping stops, the valve closes on its own and keeps beer, foam or CO2 from flowing back into the pump or gas line. The same check valve is used inside keg couplers for the same reason. If your pump pressure feels weak or inconsistent or the valve has gone stiff or brittle, that's the usual sign it needs to be replaced.