Ah it is indeed confusing as I notice, and the only explanations of the system are spread up on many random threads...
Well basically its the two game factions of Goídil and Vikings in a battle against each other. The single banners that are shown in the first post with a list of people under each are the hosts which are groups of players, each commanded by one host leader. With how things are right now only Vikingr clans that have a thread in the Clan board and at least five active members can request to form a host in these events. But for the sake of the non-clan players there is also a "public host" that is led by an experienced volunteer (in this case Kawee on the gaelic side), that everyone can apply for. The clan hosts are also allowed to invite a few guest players that are not members of their clan.
Each host has their own teamspeak (TS) channel and can agree on some things on a tactical level, like carrying a banner and in general doing the "micro-management". But there is also one strategic leader on both sides, here it is Ualraig Fer Araíg on the gaelic side and Bjorn Fenrisbarn for the vikings. These strategic leaders have no host to command on their own, but only a bodyguard for protection. Therefore they have a direct connection on teamspeak to all the host leaders and command and coordinate their cooperation, so that they can order single hosts to fulfill certain tasks.
That way it is possible to do quite a lot of tactics without a big mess if all the players of a side are in the same TS channel, and so far this system worked quite well as everyone has his place and (if the volume is turned right on ts!) they can cooperate very well.
And to participate you are best off just applying for one of the public host, for example the Dál gCais tribesmen. You don't really need to do anything to join, only on the day and time of the event itself you should join the teamspeak channel of this host so that you can hear Kawees commands, and there you will be given the password for the event server so that you can join the game and all. You don't need a microphone yourself either, as the main thing in teamspeak is that you should be able to hear your host commander and follow his orders regarding where to move and what formations to take, so that things are not just a big messy blob^^