Join a takeover of a territory during a takeover window, these are on a schedule depending on when last takeover was and can be found out on free or taken territories by clicking on a system in it and selecting the flag on the top right of the popup.
If you decide to join a territory takeover then be prepared to have numbers from your alliance to be involved, all levels should participate and have something to bring to the party.
Basic premise is to take and hold node (big blue things) in the relevant systems and hold them to earn points. Players in system also earn points, at a slower rate, but points none the less.
Pile into a given system or both depending on numbers and power, in general let the higher power ships take on opponents, and take nodes with what you have, and try to keep them. These can get pretty manic at times, so gear up your best ships with the best PVP crews you can, and other spare vessel slots with what you can. Discovery crew have some good bonuses for territory takeovers so put some of those on.
Victor is decided in 30 minute bout, whoever has the most points wins, and more points are scored near the end of the session. It is basically a numbers and power level game but the rewards are worth it.
Depending on the territory they have bonuses that apply across the alliance, including mining speeds for resources, PVP and attack bonuses for various things etc. In addition you can buy particles from the alliance store which will greatly speed up the research for territory which holds alot of good bonuses for powering up ships with Hull/Shields and other research threads. Doing this solely without holding territory is possible but alot slower as Refining these in the Territory Refinery (one at the end) is at a far lower yield.