Districts are essentially the maps that DUST 514 mercenaries fight on.
If a capsuleer in EVE Online files over a planet they can connect to a district via a defence grid satellite (doing so will make them visible to everyone in the system) and launch orbital bombardments when DUST 514 mercenaries give them targeting coordinates. The EVE ship has to be connected for 3 minutes, at which point, they can choose which targeting coordinates to fire at, and which to ignore. There is no firing cool-down beyond being connected to the satellite for another 3 minutes.
Currently, districts can be captured regardless of planetary ownership; capturing the districts of a hostile planet will make it easier to capture, conversely; holding all the districts of a friendly planet will make it harder to take over. Not all planets in the EVE Online empire lo-sec space can be attacked by DUST 514 mercenaries; only temperate planets that can support life have districts to fight over.
In the Molden Heath region, districts can be fought over and controlled by DUST 514 mercenary corporations; benefits include better rewards, ISK gained from selling surplus clones, and standard-issue bragging rights.