It is a western portion of a retaining wall that was used to keep the Temple Mount platform from spilling into the Tyropoeon Valley. The lowest parts of the wall may be from the early second temple era. The main parts are from the time when Herod rebuilt the Temple and expanded the platform – around the time of Jesus' life. Romans destroyed the Temple and most of her walls in 70AD.
Probably, this part of the wall was saved when it was covered with the debris of the upper walls and buildings during the Roman destruction. The upper parts of the current wall were built up during Moslem rule. You can see how this might have looked around 50AD. It would have been just to the right of the ramp in the lower middle.
Men must cover their heads here; even non Jews. You can use anything, but if you forget a hat, you get to use one of these cardboard hats, that are a lot like a french fry tray and don't stay on your head unless you hold them on. I always travel with a hat in my bag.
For Jews, this place is a sacred reminder of the Temple and the first Jewish nation. Many Jews will not go up to the Temple Mount itself – believing it must be purified before it is once again used for worship. So, the Western Wall is as close as they can get to the site of the Temple. You will see that the wall is segregated by gender – women and men have separate sides.
The Wall is a common place for the Bar Mitzvah, a Hebrew coming of age ceremony when boy must pray and read a portion of the Torah. Tuesday and Thursday are the most common days. Note that the women must observe from outside the men's area and often throw candy to the participants.