A byname is a surname. In particular, it is a non-hereditary surname given to an individual in order to describe him in some way.
Bynames contrast with the inherited surnames which are almost universal in the modern world.
Individual bynames, on the other hand, were the most common style of surnaming used in most of medieval Europe. Over the course of the Middle Ages, individual bynames gave way to inherited surnames, so that a man's surname was no longer a literal description except by coincidence. We've chosen to illustrate this discussion with medieval English examples, but the general principles apply to most medieval European languages. The indented paragraphs delve a little more deeply into English naming practices, and don't apply generally. An appendix lists good sources for choosing bynames from other medieval languages.