In a broader sense western wall can refer to the entire 488 metre long 1 601 ft retaining wall on the western side of the temple mount.
Temple mount and wailing wall.
The term wailing wall is not used by religious jews and increasingly not by many others who consider it derogatory.
The western wall aka the wailing wall or the kotel is the most religiously significant site in the world for the jewish people.
The first temple was destroyed by the babylonians in 587 586 bce and the second temple was destroyed by the romans in 70 ce.
For 500 years until the muslims captured jerusalem this whole southwest area where the wailing wall the kotel was known as a garbage dump.
For muslims temple mount is also known as the noble sanctuary or haram al sharif.
The wailing wall is not part of the foundation wall for the old temple it is the wall of the old roman antonia fortress.
Jewish people pray at the western wall which is structurally a retaining wall for temple mount.
The women of the new roman city built over the ruins of jerusalem were instructed by roman law to take their garbage to the dump daily wall of the old temple site.