In the process of calculating a fractal, such as the Mandelbrot set, a simple equation is applied repeatedly to each pixel in the fractal window. The equation takes the coordinates for that pixel, does some math, and produces another pair of numbers to serve as the input to the next iteration. This pair of numbers can be thought of as being a set of coordinates, and can be plotted on the fractal window.
A calculation (calc) path is simply the series of points created by iterating on one point, connected up with lines to show the movement from one to the other. The internal workings of the calculations can be made much more transparent by viewing the calc paths. Many patterns become suddenly apparent, and the differentiation between points inside the Mandelbrot set, that never escape, and points outside of the Mandelbrot set, that run for infinity, is clearly visible.