A fault can be thought of as a structural discontinuity in rock that marks movement, or displacement, to some noticeable degree. On the scale of typical topographic maps, that displacement is probably on the order of at least a few meters; if the fault has been active for a long time, the displacement can be tens to hundreds of miles. You can call the surface expression of a fault a fault line, since it’s a 2D feature. If you’re talking about its extension into the subsurface, then you’d refer to the fault plane.
Faults that are visible on the surface are mapped by geologists along with other basic geological info (rock type, other visible structures like folds, etc.). In the subsurface, fault planes can be mapped to a certain extent with the help of seismic cross-sections (or 3D seismic data, where available), and less commonly by actually drilling through the fault planes in several places. Subsurface data can help us to see the orientation of faults we know from their surface expression, as well as “blind” faults that do not reach the surface (such as the fault that caused the earthquake in India last month). There are some limitations as to how well one can see what’s going on in the subsurface, though, so we don’t have a complete map faults through the earth’s crust in any given area.
As for movement along a fault… the two sides of a fault zone (the general area around a fault plane) are not flat surfaces - think of them as each being kind of rough & bumpy. Ground up rock (called gouge) and water can help lubricate the two sides as they move past each other, but eventually the two sides will get hung up on one another. The point at which they lock up becomes the focal point for increasing stress until the rock fails - then you have an earthquake. That failure point is the focus of the quake. If there’s no impediment to movement on adjacent segments of the fault near the focus, and the energy released by the quake is large enough, then the other segments can certainly move at the same time - and you might see a surface rupture that extends for some considerable distance away from the epicenter.