The Jewish Temple in Jerusalem is gone, destroyed by the Romans in 70 AD after the Jews rebelled against Roman rule. But outside the Temple Mount, against a retaining wall, Jewish people still gather to pray.
This wall is called the "Wailing Wall" after those who stand before it to lament the loss of the Jewish Temple that once stood nearby, though it is also called the "Western Wall" because it is on the west side of the Temple Mount.
The reason for praying at the wall is that it is believed to be the closest point to the Holy of Holies, an inner most-holy room of the old Temple, the site of which is now occupied by a Muslim mosque.
The tradition of praying there apparently began around 200 or 300 AD when some rabbis taught that the presence of God still lingered at the location of the former Holy of Holies. Often Jews at the wall will write prayers on papers and press them into the cracks of the wall, toward the Holy of Holies.
The wall has unfortunately been the object of dispute between Muslims and Jews for centuries, and with increasing intensity as Jews began moving back to the area in 1800s. It remains a point of discord to this day, though the area has been under Jewish control since Israel captured Jerusalem from Jordan during the 1967 Arab-Israeli War.