A lug - but we always use the plural term, lugs - is the protrusion attached to the case where the strap attaches. Since, in most cases, the wristband is removable, the most common solution is using springbars, and thus, the lugs where the strap is attached are two on each side of the case.
Modern lugs were born when the need to wear the watch on the wrist developed in the late nineteenth century. While some people paid for newly-made cases, the most practical solution was to recover the traditional pocket watches, shorten the crown by putting it closer to the case, and weld two loops on the case to pass a leather strap, obtaining what is called today wire lugs.
But the solution was not particularly practical. So a few companies, including Cartier, lengthened the case of their wristwatches with protrusions called brancards, which were drilled to insert a small metal rod to attach the strap. Eventually, around 1925, several systems of rapid opening were invented, mainly based on springbars, which allowed a quick release of the rod to insert the strap. And the industry maintained this solution over the years since it adapted well to any watch case shape.
The industrialization of watchmaking did the rest: soon companies realized that the standardization of the distance between the two lugs, which had a parallel inner surface, was very convenient for strap suppliers, and so the current situation was standardized, where the lug-to-lug distance is used to define the size of the straps.