The patch is a client side patch. This means you'll need to download a new, albeit small, patch when you first log onto WoW today. For me, the patch downloaded and installed automatically when I executed the World of Warcraft Launcher on my desktop (pictured above).
This patch brings the game from version 188.8.131.5264 to 184.108.40.20606. Updated: The patch is 6.923mb in size on a Windows computers (interestingly the program you have to download to get the patch is an additional 2.1mb). The realms should be up by 11:00 a.m. PST / 2:00 p.m. EST today.
We'll bring you more as we learn it.