That's a small plastic housing that the small pipe goes on.
After 15-20 years it may have become brittle and then fractured. Now your car cannot pressurise the cooling system and run the higher temperature a pressurised system will run. Where this hose is should be below the 'water line'.
I think you're going to wind up replacing the radiator, as I think that barb is knackered.
You need to have that pipe off (when the coolant is cold) to start having a look to see what's what.