A guildhest is a miniture dungeon that can be entered by using the 'Duty Finder' or by taking to a Battle Warden in major cities and some smaller ones.
Most guildhests require a group of at least 4 players (2 DPS, 1 Tank and 1 Healer), later hests require a party of 8 (double the previous requirements), and should take no longer than 10 minutes to finish.
The earliest you can enter hests is level 10, you must complete enough of the story to gain access to them.
You can see the current number of completed guildhests under the 'Achievements' tab in the menu.