Trade: total credits in trade profit (including piracy, mining and trade missions) earned by squadron members.
Exploration: total credits in exploration data sold by the squadron. It's possible that the minor exploration rank increase granted by discovering new-to-you materials or driving the SRV also counts, but it'll be such a small contribution it's not worth worrying about.
Combat: based on ranking points earned by the squadron, so you get more for a Harmless squadron member killing an Elite NPC than the other way round. (Members with Elite combat rank still get ranking points for the squadron score, they just don't get any personal benefit from doing so)
Powerplay: total merits earned by the squadron for its associated power (cannot be earned without an associated power, merits earned by squadron members pledged to a different power don't count)
CQC: total CQC experience points earned by the squadron
AX: I think this one is the total value of AX combat vouchers handed in by the squadron.
Political: sum of positive actions carried out by the squadron for its associated minor faction (actions for other minor factions don't count). The way this is added up has a well-known flaw which is trivial (though very boring) to exploit, which has been acknowledged by Frontier but there is no timetable for a fix. (Cannot be earned without an associated minor faction)