The Teardops were initially designed to be a criminal syndicate - a mafia - whose goal was to accumulate power and resources in order to generate more power and resources as mafias tend to do. However, due to their distinctive modus operandi (their insistence on staying in the shadows so as not to be held accountable, trading personal favours with powerful people, and infilitrating every organisation they can) and through the sway of some players, they ended up operating a lot more like the Illuminati.
Their schemes were initially: spiking the river water with hallucinogens and then gaining a monopoly on them to make the most amount of profit; a protection racket, largely taking from thieves as a tax for not outing them; and running a group of mercenaries for hire. However, the important fact was that their leader, Afia, didn't move directly or leave any official trades or paper trails. She traded only in favours and offered people whatever she believed they wanted in order to work for her. The more resources they had the easier this was. She was quite genuinely liked. Alas, this did not save her.
Due to player action they became a lot more political. There was no end game here except to gather power. However, the more they became involved with players, the more of a conscience Afia and Matteo grew. Whilst Afia would have never been persuaded to give up her position or her luxury lifestyle, she could have been persuaded to work for the good of Monument as well, as long as she still had a level of control. She was not actively trying to hinder it when she died, but was gathering resources through harmful methods. The trouble was, she would never have agreed to work openly, as she was terrified that this would mean gambling everything she had which afforded her status and protection.
Matteo meanwhile could have been persuaded to work for the good of Monument if Afia was. She was his sole loyalty. Once she died, he was expecting to die soon after and had somewhat made peace with that.
There was a third NPC who never really took off, called Caleb, who was obsessed with power and came up with the overarching schemes on how to achieve it. Afia was responsible for actually implementing these plans through her network of friends and favours owed, and Matteo was responsible for ensuring discipline in anyone going on missions and in their military resources. However, as he never showed up much, Caleb's part was down-sized and it can be comfortably assumed he faded into obscurity with no one to implement his plans any more.
And then there was Pearl who wasn't originally a named NPC at all. Introduced partly due to give players someone to clash with in the Trident branch of the Teardrops and partly to tie in with some players' mystery quirks, Pearl was meant to only play a minor role in the Teardrop plot. However players started talking to her, quickly inviting her to session and then welcoming her into the council. Pearl valued loyalty above all other qualities and was originally loyal to Afia, before being persuaded otherwise and starting her clash with the rest of the faction.
The teardrop tattoo itself represented the idea that you were important to Afia, Matteo, or Caleb in some way. Not all people who had it were gang members and not all people who didn't weren't.