The Western Wall or the Kotel is the wall which remains standing in the Old City of Jerusalem. The wall is the only standing structure from the destruction of the second temple (Temple Mount) and has stayed fully together since 70 C.E.
The Kotel is considered the holiest spot in all of Israel. Jews will go and pray at the wall all of hours of the day and night.
At the wall, there is a divider which separates the men from the women while praying. Besides going to the Wall to daven or pray, many people will go to the Kotel to leave a prayer or note in between the limestone bricks. When you walk up to place your prayer, you will see many small pieces of paper from others. All the papers are left untouched as these are others prayers and wishes.
When you are done praying and you are ready to leave, observe how the people walk away. They never turn their back on the Wall, they walk backwards, always facing it.
Another interesting fact is all synagogues have their holy ark facing east. This would be due to the Wall and where ever you are in the world, all Jews pray toward the east.
If you cannot visit the Kotel, Aish.com has a camera where you can view the wall as well as offers a free service to place your prayer in between the bricks.