I'll take a simplified bite at trying to explain it.
Wire alignment refers to the practice of using a tensioned wire as a reference from which measurements are taken to establish straight line alignment of work pieces undergoing alignment. A horizontal tensioned wire in earth's gravity takes a well known shape called a cantenary curve in the vertical plane.
By using a steel wire of known diameter and putting a tensile load on it with a known weight a calculated tensile stress is put on the wire. From that stress and the distance between the end supports we assume we know the curve of the wire so we can take measurements from the wire to our work to determine its state of (mis)alignment.
That's what the tables Forrest posted help to do. Knowing the amount of sag at any point along the wire from an ideal horizontal straight line between the end support points lets us compensate for that sag in our measurements and thus use the wire as an ideal straight line.
Direct meaurements are taken between the stretched wire and the work piece surfaces and the tables help to reference them to an ideal straight line.
In the horizontal plane the wire projects as a straight line. For fine work it's curvature in the vertical plane is accounted for by the cosine rule in taking measurements.
What's it used for? Used to be widely used for aligning things such as bearings of large turbines and as Forrest mentioned bearings of ship propeller shafts. It was also used to align the beds of large long machine tools and locomotive frames among other uses. For some of these purposes it is still used. Steam locomotive frames are still often checked with wires. It hasn't disappeared entirely as this currrent tool offering demonstrates.
If the wire is oriented vertically it is assumed to be an ideal straight line and measurements don't have to account for sag. Sag for any wire angle between horizontal and vertical can be calculated but true horizontal wires are most often employed to simplify the method.