In opposition to the “vanilla” missions, those generated by Pirate Guild are dynamic and purely based on what's going on in your game. It means that instead of target ship spawned out of nowhere, Pirate Guild uses real ships that are really used by the game. Basically, when you do an assassination mission against a cop who dealt damages to the guild, the target has really hurt the guild and the said target was existing before the mission was displayed or accepted. Moreover the reward will be proportional to the amount of damages the said cop has done to the Guild.
As an example, when a Pirate Base decides to attack an Argon Military Centaur, a news will be displayed in the News System, and if you're in the same sector as this Pirate Base you'll also receive a broadcast message through ECS. You can decide to kill the Centaur yourself or to let the other members deal with the enemy. If you destroy the Centaur yourself, you'll get a bounty and notoriety points. Same applies with sector invasions or assassination missions. Helping pirates who are under attack will also give you money and reputation. Sometimes you may destroy a guild target without even knowing it (you'll still get the reward, though). Even if you don't want to join a fight, you may still give it a look. You certainly won't get any reputation bonus, but you'll be still able to collect some crates from the destroyed ships.
If you are a member of the Pirate Guild, nearby pirate bases may also send you a message when they are under attack. Help them destroying their foes and you'll get rewarded for each kill. (this features needs the ADS script to be installed and enabled)
You will also notice new Prisoner Military Transports and other Military TL roaming the galaxy, they are also linked to the mission system for highly rewarding but dangerous missions.
The fact the mission system is purely dynamic has both good and bad consequences:
First, and foremost, it allows Pirate Guild to deal with all kind of plugins from 3rd party scripters. If you have Race Response Fleets or Military Base Revamp or any other similar “police script” that hunt down pirates, you'll get a bunch of additional “kill the cop” missions triggered by those plugins.
Secondly, missions always exist. Apart for a few exceptions (prisoner transport), you don't need to accept them at a pirate base to complete those missions. If you stumble upon a mission target by pure luck, you'll get a reward no matter if you've accepted the mission or not (only if you're a guild member, of course). It may sound confusing to people used to the standard / click-me vanilla mission system, but once you're used to it, it's hard to go back.
The drawback is that dynamically generated missions are usually basic. I can't really put a plot or use several steps to do the said missions because it would force me to add artificial events which would defeat the whole point of my system.
That's also why the “Guild Missions” screen when talking to a Pirate Base doesn't “erase” a mission in the list when you've accepted it. It will only been deleted once you've completed the mission. Pirate Guild doesn't care if you accept a mission or not, it only cares about the result. Also notice that Pirate Guild ships and bases will try to complete some of those missions, and a target may disappear because it has been destroyed before you can reach it.
Theoretically, it also leaves room for another scripters to write plugins that will try to compete with the player at doing those missions.