Vex Axis Minds are individual Vex hulls that contain local instances of superordinate Vex goal sets. This cryptic phrase means something reasonably simple - the Axis Mind contains a copy of all the information required to pursue a particular objective.
This allows other nearby Vex to focus on their local tasks, leaving global planning to the Axis Mind. Of course, this also introduces a centralized weakness for enemies to target. But the Vex seem to consider the tradeoff worthwhile.
The Prohibitive Mind seems to coordinate Vex action against Cabal positions in Meridian Bay. The current Vex operational plan - to the extent that we assume the Vex have operational plans - appears to involve an attritional campaign on the surface, supplemented by disruptive use of gates to bypass Cabal hardpoints.