History

King Solomon constructed the First Temple at Temple Mount in 957 B.C., which was later demolished by the Babylonians. Thereafter, the Second Temple was built and then ultimately wrecked by the Romans in 70 C.E. The Western Wall is a relic of the Second Temple.

Jewish Significance

• The Temple Mount is the most sacred site in Judaism. It is believed to contain the sanctuary in which God lived on the Earth, called Holy of the Holies. Jews even refrain from setting foot on the Mount Temple so as to avoid accidentally stepping on that place, since its exact location on the Mount is still unknown. Jews believe that the place still has some Holy presence. The spiritual gravity of the place is evident from the fact that the Jews face the Temple Mount whenever they pray, no matter what corner of the world they are in.
• The Western Wall of Temple Mount is the only relic of the Second Temple that was demolished by the Romans in 70 C.E. and is considered to be the Wall located nearest to the Holy of Holies. Hence, praying before the Wall brings about greater blessings than praying elsewhere. When the Jews were exiled from Jerusalem, they would attempt to come to the Wall and pray at the possible expense of their lives.
• Jews from around the world are encouraged to pay pilgrimage to the Wall and lament over the destruction of the Holy Temple that once stood there.
• Not only do Jews physically pray before the Western Wall, but they also note their prayers on chits of paper which are folded and placed within the cracks of the Wall.

References:

https://en.wikipedia.org/wiki/Western_Wall

https://en.wikipedia.org/wiki/Second_Temple

http://www.aish.com/h/9av/j/48961906.html

https://en.wikipedia.org/wiki/Temple_Mount#Islam

https://en.wikipedia.org/wiki/Al-Aqsa_Mosque