Aetherytes are used to teleport to different locations within a City-State, and to different places within the world.
The Aetheryte system within a city is called Aethernet.
As of Patch 4.57, the main Aetherytes in the starting City-States can be used to visit other World Servers.
These massive crystals can be found throughout Eorzea, allowing adventurers to travel great distances in an instant. When first attuning to an aetheryte, adventurers will learn the spells Return and Teleport, which are necessary to use these aetherial constructs. For the ever-wandering adventurer, attuning to as many aetherytes as possible will ensure swift transport to nearly any location.
Every City-State have one main, bigger, Aetheryte and many smaller Aetherytes called Aethernet Shards, each one marking its teleport destination, while other places in the world may have just the main Aetheryte.
Free Companies with housing can place an Aetheryte housing accessory on the outside garden: when the accessory is placed, its destination is automatically added to the Teleport list.
When you attune to a main Aetheryte, its location will be added to your Teleport destination list. Attuning to the Aethernet Shard will add its location to the Aethernet destination list. When you attune to all existing Aethernet Shards of a city, your Aethernet list will get the city exit gates as new teleport destinations.
When attuned to all of the shards in each respective A Realm Reborn city-state
You now have Aethernet access to the Botanists' Guild.
You have attuned yourself to all the Aethernet shards in Gridania. The Blue Badger Gate, the Yellow Serpent Gate, and the airship landing are now accessible as Aethernet destinations.
You now have Aethernet access to the Blue Badger Gate (Central Shroud).
You now have Aethernet access to the Yellow Serpent Gate (North Shroud).
You now have Aethernet access to the airship landing.