So! It's time to discuss two things. Events, and Emergency quests. Events first...
Events is a name I've given random, well, event sorta things that pop up. These range from any number of things, to "kill this giant thing" to "kill a bunch of these little things" to the more exotic "disable these time bombs" or "dodge these bizzare alien things/lighting strikes", and even "find some keys for this door while things attack you". They add quite a bit of replayability to the missions and I quite like them. To boot they give you a meseta/exp bounty along with maybe a grinder/photon drop/synth on completion. They're good to do! Generally not super hard either, but do note that there is no penalty for just ignoring them and moving on. So if you're overwhelmed, don't be afraid to run off. They're mostly random, although there's a pattern to some (for instance, in the mission capture the nab rappies I seem to find a rockbear fight event fairly consistantly). Generally the most difficult ones are the ones where you have to fight a boss suddenly. If you remember how terrified and fucked up you got when you first fought a hildebear in PSO1, it's kinda like that. You'll die, you'll ricochet off walls, you'll feel like a little babby. But you will prevail, ultimately!
Boss fight events also have an added benefit, any weapons dropped by the boss will have an additional (boss name) soul add on on it. These add a decent chunk of stats, of varying usefulness depending on your class. Some are amazing, some are +10 HP +10 S-def.
Emergency quests! When an emergency quest happens there'll be a ship wide announcement over a PA system about some THING that's going down. Sometimes you'll have to defend a mining base, sometimes it's a large scale operation on a planet, sometimes it's dark falz trying to fuck up the ship in some manner. Most of these are excellent exp, dark falz especially, and some of them offer rare item drops exclusively from those missions. They are good to do, and they're all multi party missions, meaning they can have up to 12 people in them, like in multi party field areas, rather then the usual four. You can usually get carried through these missions, sometimes you crap out and don't get enough to fill the mission. You'll usually want to go to a semi full channel, the fuller the better, immediately upon the mission announcement going up, and rushing into the mission. From there just jump in, maybe get some 'mates, no time to waste, you'll probably get carried by some titan anyway. The later you are the less likely your team will be full and, well, good.
These are by no means mandatory, but they do provide pretty huge exp boosts and are nice to do if you're in the mood. Falz fights provide tokens to trade in for other stuff, usually just nice to have stuff like exp boosters and such. I think there are some rare weapons in there too.
So yeah! Those are events/emergency quests, do them when you see them, get prizes.