A Weave (as called by the Anagusi) or Subtle Machine (as called by the Umichooans) is non-obvious infrastructure that makes the use of Gifts possible.
In the same way that a room in a modern apartment has wires under the wall and is connected to a larger electrical network so the room responds to a press of a switch (or maybe someone clapping their hands) by toggling the light, some civilizations have taken this several steps forward and built infrastructures that give access to much more esoteric skills.
The oldest known weave that humans agree on the existence of is the Great Dream and in fact, this one seems to either be used as a base for, or be integrated in most more recent weaves.
The most obvious modern weave is the Umichooan Subtle Machine, that grants every Umichooan-blooded (mostly, there are occasional identification errors) person the ability to use Gifts.
The Portal Roads set up by the Anagusi, that respond to Dancers are yet another example of the same principle.