The Japanese used nengo (年号), eras declared by the imperial court. They didn't have a set length and a new era could be declared for any of a number of reasons: a new Emperor taking the throne, a natural disaster, astrology, etc. The longest nengo lasted for thirty-five years, but the majority were less than a decade long.

To give some a few specific examples, the Genroku era began in 1688, but when a massive earthquake struck in Genroku 16 (1703), the court declared the beginning of the Hoei era. That lasted until Emperor Nakamikado took the throne in Hoei 8 (1711), at which point it became the Shotoku era. Then when Shogun Tokugawa Ietsugu died in Shotoku 6 (1716) the Kyoho era was declared. If this sounds like a mess, you're not wrong.

I should also mention that these continue to be used in Japanese-language history books. They'll usually define an era once, telling you that Genroku 1 began in 1688, for example, and then just use the nengo.

/u/cckerberos has already mentioned era names, or 年號, but it should always be mentioned that they were originally a Chinese tradition. The era name, or nianhao in modern Mandarin, was an essential component of the imperial register in East Asian languages. For a Vietnamese or Korean monarch to take an era name independent of the Chinese one(s) was tantamount to declaring autonomy or independence from the authority of the Son of Heaven, and for a tributary state to use a certain state's era name represented subservience to that state. This was why, in the 1630s, the Manchus who had recently conquered Korea forced the Korean court to use the Manchu era name instead of the Chinese era names the Koreans had used for centuries. Era names were all meaningful and represented the circumstances of the period. For example, the first emperor of the Ming dynasty used the era name Hongwu for the rest of his life, which means something akin to "vastly martial." Like other era names this was an intentional decision, highlighting the emperor's military origins and prowess.

The era name began to be used under Emperor Wu of the Han dynasty, who reigned from 140 to 87 BC and first took the era name jianyuan, roughly meaning "establishing origins." He would take nearly a dozen era names during his long reign - as in Japan, era names were flexible and changed regularly according to the circumstances for most of Chinese history. Emperor Gaozong of the Tang dynasty used fourteen era names during his rule of 34 years!

The frequency with which era names changed was reduced greatly by the Ming dynasty, where the tradition of the emperor having only one era name for the entirety of his reign was established. Hence we generally refer to Ming and Qing emperors with their era names: the Hongwu Emperor ("the emperor of vast martiality"), the Yongle Emperor ("the emperor of perpetual happiness"), the Qianlong Emperor ("the emperor of lasting eminence").

Korea, being a weak power and often with a strong Neo-Confucian overtone, used Chinese era names. An interesting quirk: even when Beijing was captured by the Manchus, many of the Korean intelligentsia did not consider the "barbarians" to be a legitimate Chinese dynasty. This meant that there were some Koreans who used the last Ming era name, Chongzhen ("honor and auspiciousness") which began in 1627 and ended in 1644, for centuries after the Chongzhen Emperor died. In fact, there were Koreans who wrote the year 1861 as "the 234th year of Chongzhen"! Vietnam is a bit more complicated case, but the monarchs there typically used the title Hoàng Đế, or emperor, and thus independent era names.

Sources: I don't know of any good book on East Asian imperial language as a whole, and era names will generally be mentioned as a side fact in a book discussing other topics, so you might just want to check our wiki. For Korea perhaps try the article Contesting Chinese Time, Nationalizing Temporal Space: Temporal Inscription in Late Chosǒn Korea in the anthology "Time, Temporality, and Imperial Transition."