In modern Hebrew, medinah means state, a sovereign land, and is used to refer to state of Israel, Medinat Yisrael. It appears numerous times in the Book of Esther, for example (8:17): In every medinah and every city . . . there was gladness and joy among the Jews.
Medinah was originally an Aramaic word meaning the place of government, from the word din, law, with the connotation of a city. As Hebrew already had a word for city, it borrowed medinah to signify something broader, a province. Hence, the JPS translation of medinah in the above verse from Esther is in every province. Arabic has the word too; the Muslim holy city of Medina simply means city.
Medinah made it into the Yiddish lexicon, to refer to a land, not specifically a political state. America, the preferred destination for Eastern European emigrants with its famed streets paved with gold, was called Di Goldene Medine.
With the establishment of the State of Israel, medinah came to bear connotations of an independent country, not simply a province. The term mediniut was coined to mean policy and a politician is a medinai.
Referring to Israel as Medinat Yisrael (rather than, say, Eretz Yisrael, or the filthy Zionist entity) is a sure indication that the speaker is a proud Zionist.