Apart from submissions it's pretty self explanatory. Every sunday there's a set of 6 auctions with one happening each 4 hours. You bid on hats people have submitted and whoever is the last bidder/highest bidder gets it. There are 9 player hat/accessory slots and one 10th slot reserved for a special set which is usually a hat + accessory/bomb. This special item gets rotated out to something new each 2 weeks.

How submissions work is a bit iffy right now. As you might already know, you have to talk to the NPC in the room where the items get displayed. But the thing is you can only submit your item at a random time the day before the actual auction, aka saturday and only 54(?) players make it in + the window between the first and last submisson only lasts a handful of seconds since countless people wait in the auciton house to submit their items. There isn't really a notificaion or anything which tells you when you can submit your stuff so it's just luck. There was a new system being worked on which should hopefully make it better and fairer but I'm not sure if it's still a thing or how it's going.