The countries in FMA all seem to have different governmental structures. Amestris seems to be militarily run but I thought I saw references to some sort of Parliament. Xing seemed to have tribes but Ling is also the Prince. Ishval seemed to have some sort of set of elders, though I wasn't sure about that. What are the technical governmental structures of the three countries?
Short answer: Amestris is a military state which has a parliamentary republic form of government. Xing is a monarchy which has a clan structure beneath the monarch (emperor). Ishval, when it was a country of its own, had an unknown government structure but was probably split into communes; it is now under Amestris's government.
There is a long answer, too, of course...
Amestris is a Unitary State, with a population of about 50 million and a Parliamentary Republic type of Government. The Head of State is the Commander-in-Chief of the Military who holds the title of "Führer" (大総統, Daisōtō in Japanese, a form of "Generalissimo") and who concentrates the ranks of Head of State and Head of Government. — Amestris, Fullmetal Alchemist Wiki
Amestris is a military state ruled by Führer King Bradley. As mentioned above, it is a unitary state (one in which there is one central government which decides what smaller branches handle) with a parliamentary republic government. So, yes; there is a parliament, but it is more-or-less just a cover for the central military control body that we see running the country for the entirety of the show.
It is mentioned that Amestris did not used to be a military state; it is not mentioned what type of government it had prior to becoming one.
Ishval is implied to have once been a country, but is no longer a sovereign nation. Following the fall of Xerxes, many countries were assimilated into what is seen as Amestris in 1914 and onward. There is no given record of the government in Ishval at the time, though it is implied that it was a tribal land, not so much run by a government, but more likely small communes that worked on their own and were united by connections between them.
The man pictured below is a high-ranking monk, and who is most likely important in uniting the Ishval community (even prior to the War of Extermination).
Xing is a Monarchy, whose monarch holds the title of Emperor. It's composed by fifty hereditary clans all under the rule of a single Emperor, who cements his rule by taking the daughter of each clan's chief to be his concubine and bear him a single heir for each clan. — Xing, Fullmetal Alchemist Wiki
The country of Xing is, as stated above, a monarchy ruled by an emperor (part of which makes it an empire). Beneath the (unnamed) emperor are approximately fifty clans, most of which have an heir (about forty-three have heirs, the others do not). Two of the clans we are briefly told about are the Yao and Chang clans.
When Ling says he is a prince, this is basically a technicality; he is the leader of the Yao clan and an heir to the emperor's throne (the twelfth heir, I think). He has no special privileges above being the leader of his clan, so his rank as "prince" solely has to do with his position as heir.
Drachma, the large and arctic country to the north, is not told to have any specific kind of government. However, given how it is the largest country around and is an arctic country, as well as the types of wars it fights, it is implied that it is an equivalent of Russia. Russia was, at the time, the Russian Empire, which I believe was also ruled by an emperor or tsar. Further reading: (Russian Empire: Government and administration, Wikipedia)
Creta, to the southwest, is a smaller country which is composed of tribes in a federation-like manner. (Further reading: Creta, Fullmetal Alchemist Wiki)