Several parts combine into one character.

府 is a phonetic-semantic compound: 广 (the "sloped roof" radical) + 付 (fù, "to give / hand over" — providing the sound). The composite reads as "the building where one carries out received orders" = a bureau, the seat of an institution. From this institutional building came the abstract sense "official seat / location of authority / prefecture."

Korean reading "bu." 政府 (jeongbu, government — universal political vocabulary), 官府 (gwanbu, government office), 司法府 (sabeopbu, judiciary branch — Korea's three branches of government are 입법부 ipbeopbu, 행정부 haengjeongbu, 사법부 sabeopbu), 樂府 (akbu, "music bureau" — historical Chinese-style poetry collected by ancient music bureaus, used as a literary genre name).

Mandarin fǔ, 3rd tone. 政府 (zhèngfǔ, government), 总统府 (zǒngtǒngfǔ, presidential office — Taiwan's presidential building is the 总统府), 王府 (wángfǔ, royal mansion — historic palaces of imperial princes; 王府井 Wángfǔjǐng is a famous Beijing shopping street named for a former princely residence with a well).

Japanese on-reading フ (fu) — 府 (fu, prefecture / bureau), 政府 (seifu, government), and uniquely Japanese: 府 names a specific level of administrative subdivision. Japan's prefectures break into four types: 都 (to, Tokyo metropolis), 道 (dō, Hokkaido), 府 (fu, only two: 京都府 Kyōto-fu and 大阪府 Ōsaka-fu), and 県 (ken, the standard prefecture form, 43 of them). The 都道府県 (todōfuken) system gives Japan's 47 prefectures four different name-types based on historical and political significance. Kyoto and Osaka retain "fu" status from their pre-modern roles as historical capitals and major commercial centers.

Memory aid: a sloped roof (广) over giving / handing over (付) — the building where orders are received and executed = bureau / prefecture.