Hope Island is where Gaia resides. After she calls the Planeteers, they live on Hope Island with her.
Hope Island is apparently sustainable - the Planeteeers live there between missions, in separate sleeping quarters. There are gardens and beaches, which the Planeteers are responsible for maintaining - though their duties are usually taken to and seen as hobbies, rather than chores.
The Crystal Chamber on Hope Island is where the Planeteers meet before missions, usually gaining information from Gaia and the Planet Vision screens.
In episodes such as Heat Wave, it is learned that if Hope Island comes under attack and dies, Gaia dies as well.
The exact location of Hope Island is never revealed in canon, though executive producer Barbara Pyle revealed during the 2012 Reddit IAMA that Hope Island is near the Bahamas.
The Crystal Chamber
The Crystal Chamber is where the Planeteers gather to be briefed by Gaia before each mission.