A fault is a large crack in the Earth's crust where one part of the crust has moved against another part. This movement means that faults prove the Earth is an active place. They are signs of powerful forces deep underground.
The parts of a fault are (1) the fault plane, (2) the fault trace, (3) the hanging wall and (4) the footwall. The fault plane is is a flat surface that may be vertical or sloping. The line it makes on the Earth's surface is the fault trace. Where the fault plane is sloping, the upper side is the hanging wall and the lower side is the footwall. When the fault plane is vertical, there is no hanging wall or footwall.
Any fault plane can be completely described with two measurements: its strike and its dip. The strike is the direction of the fault trace on the Earth's surface. The dip is the measurement of how steeply the fault plane slopes—if you dropped a marble on the fault plane, it would roll exactly down the direction of dip.
U.S. Geological Survey image
It's important to know a fault's type: normal, reverse or strike-slip. The type reflects the kind of forces that are acting on the fault.
Normal faults form when the hanging wall drops down. The forces that create normal faults are pulling the sides apart, or extensional.
Reverse faults form when the hanging wall moves up. The forces creating reverse faults are compressional, pushing the sides together.
Together, normal and reverse faults are called dip-slip faults, because the movement on them occurs along the dip direction—either down or up, respectively.
Strike-slip faults have walls that move sideways, not up or down. That is, the slip occurs along the strike, not up or down the dip. In these faults the fault plane is usually vertical, so there is no hanging wall or footwall. The forces creating these faults are lateral or horizontal, carrying the sides past each other.
Strike-slip faults are either right-lateral or left-lateral. That means someone standing near the fault trace and looking across it would see the far side move to the right or to the left, respectively. The one in the picture is left-lateral.
In reality, many faults show a combination of dip-slip and strike-slip motion. Geologists use more sophisticated measurements to analyze these fault movements. The Natural Fractures site has a page with more rigorous detail on these.
You can judge a fault's type from looking at the focal mechanism diagrams of earthquakes that occur on it—those are the "beachball" symbols you'll often see on earthquake sites.