The Guild leader realm transfer service (called "Guild Master Realm Transfer" by Blizzard) is part of Guild Master Services under their overall Guild Services support (although they are basically the same thing). It allows a guild leader to transfer the realm of the guild and the guild leader character at the same time for a one time fee of US$35 or €30 (£25). A guild name change is free with this service. Combining this service with a guild leader faction change gives a discount of US$10 (transfer US$35 + faction change US$40 - US$10 = US$65 total) or €10 (transfer €30 + faction change €35 - €10 = €55 total; £5 discount, transfer £25 + faction change £30 - £5 = £50 total).
When the guild leader changes realms, the framework of the guild goes with the guild leader as well, including the guild vault, level, perks, and achievements.
The original guild will continue to exist, but it's guild level will be reset to 1 and will not retain its guild bank or any of its achievements, reputation or progression. The guild leader initiating the realm transfer will need to choose a new guild leader for the original guild. Various requirements and restrictions apply to the guild leader and the appointee to lead the original guild.