Wires are the black lines that can be drawn on top of, or instead of, the surface of an equation (depending on the render mode).

Wires are actually OpenGL lines rendered in 3-space. The wire width is the width of the wires in some small arbitrary units.

The Plot wire every... option is provided for those situations when you want many x/y or r/t intervals for a nice smooth graph, but you don't want as many wires. This option can also help in placing wires at specific intervals (eg. if you are plotting a 10 x 10 unit region using 100 intervals in each direction, you could set this option to 10 to plot a wire every 1 unit).