Hi
As the others say, the black and brown part is the electrical actuator for the EGR valve. The EGR valve disc is in the metal part on the left.
Next to the EGR valve is a long "box" shape metal part. This is the EGR cooler. Engine coolant flows through an outer jacket, to remove heat from the recirculated gas. There are coolant hoses attached, check these for leaks.
I don't know if there is a "cooling water section" in the EGR valve also. If there is, check its hoses for leaks. You may might need a new gasket at the joint between the brown (plastic ?) part and the metal part, but I do not know if there is water at that point.