The underlying implementation of graphics in the platform uses Bezier curves, named after the man who first made use of them to design car bodies. Why do we use them here? Because the world has adopted these annoying-but-useful little curves; and all drawing packages, CAD packages and design software understands them, so there are a lot of people who have been trained in their use. Are there better solutions? Well, yes, but let’s not get into that right now, lest we upset the Math Police.
The idea of a Bezier curve is fairly simple. You have a start point and an end point, and a curved line that runs in between them. If you have a series of these, you can draw any kind of shape. The fun part is that the curve between the two endpoints is controlled by two ‘Handles,’ one connected to each node. In a simplified way, the first handle (which is connected to the start point) will determine how the line will curve as it takes off from the start. Likewise, the second handle will determine how the line comes in to the end point.
The benefit to embroiderers using these curves is scalability; Size does not matter. So if you make a Run stitch object using a curve, those stitches will generate nicely on that path at virtually any size, and the sewn result will be a nice clean curve even when made large.