A condominium is a type of non-sovereign polity where ultimate governance is assumed by two or more foreign countries. This status is most commonly chosen by the League of nation to avoid conflict from multiple claimant to a given mandate.
Condominium status can take more than one form enumerated below.
Top Most Condominium
The country is autonomous all internal aspect but the head of state (and probably head of ministries and government agency) needs to be accepted by both mandatories. Alternatively, some form of elected parliament submit a list of candidate to both mandatories who chose one from among it.
Inhabitants have originally only the citizenship of the condominium but are probably able to gain one of the mandatory's one more easily and/or can get a special type of visa to work and study in the mandatory.
This is probably best for buffer states. It doesn't cost anything to maintain (due to their autonomy) and the mandatory confirmation of head of state ensure that it won't be one that favour one mandatory over the other.
One example of this is the Aaland Island
The condominium is divided into 2 or more enclaves. Each one is considered a part of one mandatory and its inhabitant have its citizenship. There is a central government with very limited powers (mostly things like ensuring the maintenance of roads, postal services, etc...). If there is a great number of enclaves spread all over the place, giving the land condominium status could have been the best way to avoid starting (or more probably, restarting) a war over borders. The confederation is probably not considered a distinct country by other governments and organisations (no Olympic delegation or embassy for example).
Part of the country is controlled by 2 or more mandatories. The parts in question might be contiguous with an mandatory or might be inhabited by an ethnic group related to it. The central government is sovereign except is certain matters within those zones. These matters could be cultural (the curriculum on a zone being dictated by a given mandatory) or physical (no engineering work is allowed in the zone without the mandatory permission).
This type of condominium could be a more forceful type of "top most condominium" described above. Alternatively, it could be an occupied country on its way toward recovering autonomy: the government is sovereign over the country except for matters regarding the demilitarized zones near the mandatories' borders.
There is no central government but rather, decision are taken by consensus between the local representatives of the mandatories. Various regions could be administered by unelected representative that answer to one mandatory (depending on the situation, military governor, company officials, etc...) or they may alternate governance between them.
If there is (or was) a native population, its right and obligations might vary from one region to another where they may or may not have some form of representative. mandatories citizens might be "above the law" in regard to the native population but would still have to answer to his own government (there might be an agreement that citizens must be tried by their own mandatories no matter where a crime was committed).
This is the most unstable of condominium and few, if any, modern ones still exist.