Kinjo Tenno (the present emperor) (今上天皇)
Kinjo Tenno (the present emperor) is the name indicating the person who is on the throne at the time.
The present emperor as of June 2010 is the first Prince of Emperor Showa and his imina (personal name) is Akihito (the 125th Emperor).
Regarding the naming of Kinjo Tenno (the present Emperor)
Kinjo (今上) is the kanji (Chinese characters) representing 'imanoue' of yamato-kotoba (words of Japanese origin) and is the word meaning the current mikado (emperor) as with seijo (聖上) derived from kango (words of Chinese origin). It is also called togin (the current Emperor).
In Japanese culture, there is no specific name to indicate the persons to be treated with respect and thus the name for the present emperor has not been developed well. However, Kinjo Tenno (the present Emperor) has been increasingly used in expressions in which neutrality is required because, when writing it along with Emperor Taisho and Emperor Showa, the expression of 'Kinjo' or 'Kinjo Heika' (the reigning emperor/His Majesty the Emperor) doesn't feel right and the expression of 'Kinjo Tenno' can be recognized as an objective expression. Moreover, Empress Michiko calls him 'Kinjo Heika' in public.
Though 'heika' (His Imperial Majesty) is used as the title of honor for the emperor (prescribed by Imperial House Act) as with kings and queens of foreign countries, the expression of Kinjo Tenno Heika is not used and the emperor is only called Kinjo Heika, Tenno Heika or Heika, or Mikado which is rarely used now. Also, the names of Emperor Meiji, Emperor Taisho, Emperor Showa and so on are posthumous titles which denote respect by themselves and therefore the expression of Showa Tenno Heika is not used (Orally 'Showa no Tenno Heika' (his Imperial Majesty of Showa) may be used and in this case 'Showa' is construed as 'Showa era'. However, the Empress Michiko calls the Emperor Showa 'Sentei Heika' (His Imperial Late Majesty) in public).
Example at the present time
At present, in Japan, the name of 'Kinjo Tenno' (the present Emperor) is not often used in spoken language. Public institutions such as government and major media often call the emperor 'Tenno Heika' (his Imperial Majesty) with Heika which is the title of honor prescribed by the Imperial House Law. Maybe as a result of this, most people usually use the name of 'Tenno Heika'. On the other hand, persons who are critical of the so-called 'Emperor system' and the academic world which basically tends to avoid the title of honor and so on often call 'Tenno' (the Emperor) without an honorific title.
When multiple emperors become the subject of conversation about history, the expression of 'ima no Tenno' or 'ima no Tenno Heika' (the Emperor of the present) may be orally used.
Moreover, since tenno is called 'The Emperor' in foreign countries, there is an example of 'name + Tenno' (for example, 'Hirohito Tenno' (the Emperor Hirohito) in the movie "The Last Emperor"). Similar examples can be seen even in Japan, but calling the emperor by a name without an honorific title can be often perceived as a declaration of intention that special respect is not paid to the emperor.
Regarding the naming of Heisei Tenno (the Emperor Heisei)
In some publications and printed matters, 'Heisei Tenno' (the Emperor Heisei) is used as the third person without using the title of honor and there are arguments for and against this.
It is said that such notation came from the fact that Emperor Showa was already expressed as 'Showa Tenno' (the Emperor Showa) instead of 'Kinjo Tenno' (the present Emperor) in some publications after World War II.
The name 'Heisei Tenno' (Emperor Heisei) is the form of posthumous title or imina (posthumous name) of the person concerned and therefore it is incorrect to use it during his lifetime. A posthumous title is the title given after the demise of the Emperor and gengo (era name) does not necessarily become his posthumous title from an institutional standpoint (after Meiji period the custom was just followed and nothing is prescribed for the posthumous title in Kyu Koshitsu-rei (the Old Imperial Families' Act) and so on).
There are many people, particularly among the so-called conservatives, who think that the name of 'Heisei Tenno' (the Emperor Heisei) is not used on television or in the newspaper because it arises from the idea that 'it is irreverent to use "Heisei Tenno" (the Emperor Heisei) which should be a posthumous title of the present Emperor during his lifetime' as described above.
The name 'Heisei Tenno' (Emperor Heisei) is not considered as a wrong expression and there are celebrities including Ichiro FURUTACHI who usually use the name of 'Heisei Tenno'. It is convenient to call 'Heisei Tenno' (Emperor Heisei) in the meaning of creating uniformity in the names of the emperors or following the rule of 'name of gengo (era name) emperor' (the practice of assigning one era name to one emperor).