Reigning Emperor translates the term kinjō tennō (今上天皇), or Majesty translates the term heika (陛下) and according to protocol, this is how one is supposed to refer to the Emperor of Japan now currently living, in order to avoid calling him by his title of "Emperor Meiji", or "Emperor Taishō" or "Emperor Shōwa" (i.e., Hirohito) while these sovereigns were still living.
This is because in the title "Emperor Showa", the "Showa" portion is called the emperor's gō or more specifically shigō (諡号), which is the name that is given or "sent" (okuru (諡る)).
Although in the West, it has become customary to refer to the sovereign by his given name, such as Emperor Hirohito, this is not done by the Japanese. The only context where his given name is used is when referring to his time as Prince Hirohito (裕仁親王).
One should also note that members of the Imperial household have additional -miya type titles. The current Crown Prince was usually referred to by this title, "Hiro-no-miya" before the death of his grandfather.
History
Under the modern system, the gō title of the emperor will always match the era name (regnal year name; gengō (元号)). However, this issei ichigen system (ja:一世一元, "one generation one title") was only implemented in the modern age of Meiji Restoration.
In the past, the emperor's name never matched the regnal era name, and the change of the regnal era name (kaigen) could occur any number of times.
Additionally, some emperors had two gō titles, when reoccupying the throne in a process called chōso (ja:重祚). One example of this was Empress Kōgyoku, who later mounted the throne as Empress Saimei.