Emperor
In present-day English, the "emperor of Japan" refers to the tennô 天皇, the monarch who has reigned over (but not necessarily ruled) Japan for virtually all of recorded history. It is uses to refer to the tennô of any period of history. However, this English term apparently was not used for the tennô until sometime after Perry's visit in 1853.
The 16th-century Jesuits when describing the tennô used terms like "nobleman," "king," "prince," but not particularly "emperor," if at all. When referring to him, they usually called him the dairi 内裏, a word that originally referred to the imperial palace, but had come to be used as a title for the tennô, similarly to the title Pharaoh. (The "emperor" doll in the Doll Festival set is still called the dairi.)
The Jesuits also called the daimyô kings, which was hardly a misuse of the European term, as the daimyô, especially of Kyushu where the Jesuits were first active, were independent rulers who really ruled their territory and fought each other. The tennô was described as the hereditary ruler of the whole country, but one who was no longer obeyed.
Will Adams, an Englishman who arrived in Japan in 1600, referred to Ieyasu, who became the first Tokugawa shogun in 1603, as "king" in a 1611 letter to his wife, and as "emperor" in a pamphlet of the same date. "Emperor" in Europe referred to someone who ruled over kings, so in the latter he was clearly indicating that Ieyasu ruled over all the various "kings" that Europeans knew existed in Japan. The Englishmen who arrived in England with the East-India Company in 1613, also referred to the then retired shogun Ieyasu and the shogun Hidetada as "emperor," both in public and private, so using this term for the shogun seems to really have taken hold among the English, and this use presumably was taken over by the Dutch.
In 1616 Richard Cocks mentioned the "dairi," and Kaempfer, who came to the Dutch settlement in 1690, referred to the tennô as the "Ecclesiastical Hereditary Monarchs," so it seems the English and Dutch never used "emperor" for the tennô.
One sometimes comes across statements to the effect that during the Edo period the secretive Japanese told the Dutch that the shogun was the emperor, hiding the existence of the real emperor, but such statements are made in ignorance of early Japanese-European contact. It is clear that the Europeans, though they knew all about the tennô, chose "emperor" as the word most suitable to describe the shogun.
Reference: They Came to Japan