Short answers, pulled from the story.
Miklós Horthy was the Regent of Hungary from 1920 to 1944, ruling a constitutional monarchy that had no actual king. He was the last commanding admiral of the Austro-Hungarian Navy and was chosen as regent on the 1st of March 1920 after right-wing forces restored the kingdom and blocked any Habsburg return.
The Treaty of Trianon, signed on the 4th of June 1920, reduced Hungary's territory so substantially that the country lost over 70 percent of its historic lands. Romania took Transylvania, Czechoslovakia received much of Upper Hungary, and Vojvodina went to the Kingdom of Serbs, Croats, and Slovenes, leaving more than three million Hungarians outside Hungary's new borders.
The Arrow Cross Party was an extremist, antisemitic, and pro-Nazi movement led by Ferenc Szálasi. Germany installed Szálasi as Hungary's leader in October 1944 during Operation Panzerfaust after Horthy attempted to negotiate an armistice with the Allies. Szálasi's militias were responsible for the murder of between 10,000 and 15,000 Hungarian Jews.
The monarchy was formally abolished on the 1st of February 1946, when the Second Hungarian Republic was proclaimed. That republic was itself short-lived and was succeeded by the communist Hungarian People's Republic in 1949.
The First Vienna Award reassigned the southern parts of Czechoslovakia to Hungary following the Munich Agreement. The Second Vienna Award, arbitrated by Germany and Italy, assigned Northern Transylvania to Hungary from Romania. Hungary also occupied and annexed the Carpatho-Ukraine after Czechoslovakia was abolished.
The loss of territory under the Treaty of Trianon stripped Hungary of agricultural and industrial areas, forcing reliance on exports from its remaining farmland. The Great Depression accelerated the collapse from 1929, partly due to the failure of the Österreichische Creditanstalt bank in Vienna. From the mid-1930s onward, Hungary's trade became deeply dependent on Nazi Germany's economy.