That depends on the nation.

The one I have most thought out at the moment is the Valiero'ji. Each country in the nation has a ruling family. The Head of House of each ruling family is effectively the monarch of their state. Every five years there's a national election, where one of the stately houses will be elected royal family of the Valiero'ji. The Head of House of said family will then take the role of national monarch and move to the capitol, Valdin, with his or her immediate family (spouse and children, if he or she has any (I say "she" as well, because the Head of House in Valieren society will always be the eldest heir, regardless of gender)). The ruling of the family's country (Stewardship) will then fall to the second heir or the family, be that the monarch's younger sibling or his/her parent, cousin and so on. What concerns law, each country can formulate their own laws, as long as they coincide with the nation's constitution and some ground laws for economy and civil services. Just for interests sake, the current monarch is King Argos Helia Ahron, of House Ahron. House Ahron rules over the state of Ravan, which is a huge valley and also the only very horizontal Valieren state (yellow state) on the map in my above post. The current Steward of Ravan is Argos' younger brother, Thale Helia Ahron, in his stead.

The Svia'ji have a diarchy ruling the whole nation and seperate diarchies ruling each of their respective countries.

The Aari'ji is very similar to the USA, having an elected president or prime minister, which each of the countries effectively acting as federal states. Some of their states are more like colonial territories, though, but the same rules apply.

The Iga'ji has a governor for each of its respective states, but only for administrative purposes as the entire empire must adhere to the laws of its emperor.

The Col'tun'ji is the most simple, literally being a union between multiple separate tribal clans on the southern plains. Each clan can basically do whatever they want, but must answer to the Supreme Chieftain/ess - who is effective chosen among the children of the whole nation during the previous Chief's prime by a bunch of elders who can apparently recognize the required traits for leadership in a young child.

The Arah'ji is similar to the Arab emirates in rule, which each state having a lord (that can come to power in whatever manner works for the people of that state) who answers to a High Lord at the capitol. All their laws are based on a fairly strict religion as well, so laws are pretty much taken for granted in each state. At one point they were made up of your "feudal states", while under Iga'ji rule, before the Aari'ji effectively freed them after a World War (the Iga'ji isn't made up for those feudal states anymore because of the egregious strictness and power of their current emperor).

I haven't really thought of the Tita'ji or Qui'a'ji's laws yet. I expect the Qui'a'ji to be pretty much like the Col'tun'ji (but less violent) and the Tita'ji I'm thinking will have a Kingdom of Iceland type structure ruling over a generally Inuit-like populace. Not sure yet.