It's mining, missioning, roams into low, null, and WHs, and exploration. Uni activities don't neatly fit into their respective campuses and each of the campuses don't confine their activities to one type.
Uni HQ is historically Aldrat, which is a system in a dead-end pocket close to Hek... But, the thing is, Alrat actually sort of sucks. It might provide decent isolated mining or missioning, but it is far from high-reward content, so basically all of the unistas which still lived in high sec and were actually involved in the community decided to pick up and move to a system that was closer to the low-sec campus, and also close to Stacmon, out of which Chill Town is based (a prior, similar effort). The HSG is set up to be close to null and low, without being resource-scarce island life... That said, it isn't really in any particularly good way sanctioned by Uni leadership, so it doesn't have status as its own campus.
To go through all of them:
HSG is as described above. High sec players, a mix of pvpers and pure carebears, including industry people, miners, null sec explorers, etc locate on a system towards the edge of high sec.
LSC, NSC, and WHC are fairly self-explanatory for Low Sec, Null Sec, and Wormhole Campuses, teaching players how to live in their respective security spaces. It's notable that wanting to do low-sec pvp does not mean you're instantly a LSC member, etc. Uni campuses are these separate things that members have to apply to join, with each having its own requirements. Campus members typically are going to be in separate LSC channels in mumble and in game for most of their interactions, so it's sort of like a corp within a corp. People in the Wormhole campus are only occasionally going to ever actually interact with members of the High Security group, for instance, because they mostly show up to different fleets and spend time in different Mumble channels.
Project Solitude is living out of Solitude, as one might expect. It's population seems to be fairly small lately, as many people who have tried it out abandon the idea. Island life provides shelter from war targets, and a pretty different dynamic from any other type of space... But the problem is that this type of space is rather boring. Exploration sort of sucks around Solitude IMO because most of the other people living there are explorers. Sites tend not to last long, and then aren't particularly rewarding because most of the surrounding area is low, rather than null. The solitude resident must also buy and sell at island prices, go into low if they want L4 missions, and, if looking for pvp is presented with a rather sparsely-populated region with mostly explorers... There is very little going for Solitude in terms of actual content IMO.
The incursion community is an incursion community. It's a pretty great one for learning the ropes of incursions, and is okay with sub-optimal ships/fits if using them helps newer players learn incursion (and fund more expensive ships)... Part of it is that they normally seem so starved of people that they can't form fleets though. They rarely seem to actually get much new blood though, so while they're rather all-accepting, it also seems to be pretty much all veteran players with very blinged ships, and alts gallore for scouting/boosting/picketing/hacking/chauffeuring/hauling.
Amarr Mining Campus is a super-secret system that unistas mine out of that pretty much anyone could find if they invested any effort at all. Yeah, it is mostly miners, but they also run defense fleets or something I guess.