Round thing on rear brake line right side 1999 concorde
 

https://cimg4.ibsrv.net/gimg/www.chr...f116382b4.jpeg
What is this part called? One on each side from solid line to flex line to caliper. Mine is leaking. Can’t seem to find a part number. What exactly does it do? 1999 Chrysler Concorde

I have not worked with your particular model vehicle. But from the picture you supplied here, those would be Brake Proportioning Valves.