"Wailing Wall" is a term used by many non-Jews and some less traditional
Jews to refer to the only standing remainder of the
Second Temple, destroyed by the
Romans in 70 A.D. (or C.E., as many observant Jews measure time). Many Jews consider "Wailing Wall" to be a slightly offensive term at best.
The term was derived from what these non-Jews assumed to be mourning: prayers made by pilgrims to the Wall. These pilgrims are no longer the most common visitors to the site, as Orthodox and especially Ultra-Orthodox Israeli Jews pray there everyday. While some prayers (of the pilgrims and pious alike) may be made in or from grief, many more are simply heartfelt prayers to God. For true mourning of the Temples, one should visit on Tisha B'Av, a fast day which does, in fact, remember the destruction of both the First and Second Temples.
Today, many Jews simply refer to the Wall as "the Wall"; some refer to it as "the Western Wall." Those who speak of it in Hebrew call it "Kotel Ha-Ma'aravi," which translates to "Western Wall."
The Wall was once part of an outer wall surrounding the Temple Mount, where the Temples were built. This is traditionally the holiest spot in Judaism; ironically, the Wall itself was not a part of the Temple, and thus held no religious significance until the Second Temple's destruction. Somewhere upon the Mount is the original location of "the Holy of Holies," -- the room in the First Temple where the two Tablets of the Ten Commandments were kept. Only the High Priest was allowed to enter that room, and he for one day a year only: Yom Kippur, the Day of Atonement. Some believe that Jews are forbidden to visit the Mount, for fear of accidentally standing in the location of the Holy of Holies, and inviting God's wrath. This ignores the fact that Al-Aqsa Mosque and the Dome of the Rock stand upon the Temple site, and so far God has held his wrath in check.
The Wall fell under Jordanian jurisdiction after Israel's creation and War of Independence in 1948. Although a 1949 armistice between Israel and Jordan allowed for visits to the site by Jews, Israeli Jews were in fact prohibited from traveling to the site until 1967, when it was captured by the Israelis during the Six-Day War. Moshe Dayan is believed to have been the first to revive the tradition of inserting written prayers in the cracks of the Wall; his own was for peace for the "House of Israel."