Evidently the patch install is messing up things for a lot of people. If you're having trouble, refer to this thread in Tech Support forum:


It helped me get logged in finally though I still had to jump through hoops to get to the realm selection screen.

Basically, the steps for Windows users that worked for me to fix it were:
Close out the launcher.
Open Task Manager and make sure that agent.exe isn't running under Processes. End it if it is.
Open the ProgramData folder and delete the Battle.net folder.
Open the WoW folder and go to the Updates folder, delete the file that includes 16357 in the name.
Restart the launcher. You'll have to wait a few moments for setup files to update again then the patch to download.

Once I logged in, I got the generic pick a realm type screen, got dumped into character creation on a random realm, and had to back out to get the normal realm selection screen.