Normal faults
where the footwall moves downwards (i.e. towards the centre of the Earth) relative to the hangingwall.
Reverse (or thrust) faults
where the footwall moves upwards (i.e. away from the centre of the Earth) relative to the hangingwall.
Schematic illustration of normal and reverse faults. Note that the view is a cross-section through the Earth, such that the up-direction on the page is away from the centre of the Earth.
Strike-slip faults
where the fault surface is vertical and the footwall moves either left or right (with respect to the plane perpendicular to the fault and to the Earth's surface). Strike-slip faults with left-lateral motion are also known as sinistral faults. Those with right-lateral motion are also known as dextral faults.
Schematic illustration of the two strike-slip fault types. The view is of the Earth's surface as from space.
Geological fault refers literally to a crack in the rock of the Earth's crust. Such faults are the source of many earthquakes that are caused by slippage vertically or laterally at the fault. The largest examples are at tectonic plate boundaries. These are not exclusive. Many small faults are known to exist that are far from active plate boundaries.