Advantage is a faction warfare mechanism that increases your victory points per complex capture. Without it, hundreds of extra complexes are needed to capture a system, so advantage is critical especially if both sides are trying to capture a system.
Both sides in the war can build advantage in a system, and the "net" advantage is calculated by simply subtracting one side's advantage from the other's. If Minmatar have 58% advantage an Amarr have 36% then the result will be 22% net advantage for Minmatar.
Advantage is calculated using the actions in the system itself, and in neighbouring systems, and has a maximum value.
To gain advantage in a system...
To reduce the enemy's advantage...
Exploration in enemy rearguard systems yields essential components for structures, in particular the listening outposts that provide advantage. The sites you are looking for are Operations Centers, which drop Amarr Encoder Splices and Amarr Decoder Packages.
These sites are found by probing down cosmic signatures in enemy rearguard systems, and then hacking the correct can with a data analyser. There will be multiple cans per site, at most one will have what you are looking for, and NPCs will spawn and attack you as soon as you finish the first. The best approach is therefore to use a cargo scanner to find which can to hack, and align out before looting it.
The Encoder Splices and Decoder Packages can then be traded (along with some loyalty points) for Propaganda Broadcast Structures or Listening Outpost Structure at a Tribal Liberation Force LP store.
These sites spawn randomly and provide a 2% advantage bonus for the relevant faction while they exist. They are defended by NPCs and have a reasonable amount of HP. When they die, all characters that did damage to them get 15,000 LP.
To avoid being killed or driven away by the NPCs, either kill them before shooting the cache/depot, or have enough active tank (EM/Therm) to ignore them.
Winning battlefields gives 15% advantage, equivalent to seven and a half rendezvous points.
These are very relaxed (if done correctly) combat sites that reward loyalty points and advantage points for completion. They are cosmic signature sites that must be probed down, but are very easy to probe; they always spawn within 8 AU of a celstial and can be probed to 100% with an 8 AU scan.
Each Rendezvous side provides 10,000 loyalty points and roughly 2% advantage. They can be completed in around ten minutes, and respawn after about 30 minutes.
With the AFK Vexor, warp in to the site, launch drones, set them to aggressive, and orbit warp-in while letting the drones kill the rats. Once the officer battleship is dead, warp out.
[Vexor, Amarr Rendezvous]
Medium Armor Repairer II
Drone Damage Amplifier II
Drone Damage Amplifier II
Drone Damage Amplifier II
Medium Armor Repairer II
Large Compact Pb-Acid Cap Battery
Cap Recharger II
Cap Recharger II
10MN Afterburner II
Core Probe Launcher I, Core Scanner Probe I
Drone Link Augmentor I
Drone Link Augmentor I
Prototype Cloaking Device I
Medium EM Armor Reinforcer I
Medium Thermal Armor Reinforcer I
Medium Thermal Armor Reinforcer I
Hammerhead II x5
Core Scanner Probe I x8
For a more active approach, try the autocannon Wolf guide.