World of Warcraft, like most games, throws up several error codes if something goes wrong like the game crashing. In this case, WOW51900101 is some login error, and it tends to be widespread.
This error has cropped up a few times over the years, sometimes with benign causes and sometimes not. Supposedly when Blizzard has been DDOS’d (Distributed Denial of Service; a cyber-attack which prevents access to an online service) in the past, this same error message has appeared. However, it also sometimes happens before or after server maintenance or some patch.
There are some fixes you can potentially try; some users report that clearing your WoW cache or uninstalling and reinstalling the game work, but others report those methods failing to resolve the problem. The issue with this error is almost certainly server-side, and so will need to be fixed on Blizzard’s end whenever it appears for most users, though some may get lucky and occasionally bypass whatever the real problem is when attempting to log in.