I mean in terms of years (though, for that matter, I'd be interested in the months too; obviously traditional Chinese festivals are still based on the Lunar calendar, which is why the New Year moves a bit every year, but most people still use a 31 day Yi-Yue, 28 day Er-Yue, 31 day San-Yue, 30 day Si-Yue, etc).
The calendar was extremely important in pre-modern China. What is now called the lunar calendar (nónglì 農曆 "agricultural calendar") in English was actually a lunisolar calendar, and it's use can be traced back to the Shang dynasty (c. 1600 BCE–c. 1046 BCE) (where it was called the yinyang — here meaning lunisolar — calendar 陰陽曆). Setting the calendar was something that only the ruler had the right to do. Later dynasties would strictly control the production of calendars and private ownership of astronomical tools was forbidden. The reason for this was that legitimacy of rule came from heaven — a word tiān 天 that also means "sky" — and the movement of the heavenly bodies were interpreted as signs of heaven's approval or disapproval. There is even evidence on the ancient "oracle bones" (jiǎgǔ wén 甲骨文) of astronomical measurements of comets and eclipses. The calendar was very much a sign of dynastic rule.
The system used to mark the years is usually referred to as the sexagenary cycle in English, but would be more literally translated as the "stems and branches" (gānzhī 干支). Basically there are ten "heavenly stems" (tiāngān 天干) and twelve "earthly branches" (dìzhī 地支). The heavenly stems were the names of the ten days of the week during the Shang dynasty. The earthly branches were the names for stages in the orbit of Jupiter. They were used as ordinal numbers. To record years, you would take one heavenly stem character and one earthly branch character and put them together. So the first year would be 甲子, because jiǎ 甲 is the first heavenly stem and zǐ 子 is the first earthly branch (these characters have other meanings, but in this context those meaning are irrelevant). The effect is like saying A1, as both A and 1 are from different systems and in this context both mean the first. This system of counting wasn't only used for years, it was used for days and hours as well (though never for mathematics). Also, the earthly branches are used to refer to the twelve signs of the Chinese zodiac.
This system made for a 60 year cycle, whereupon it reset and went back to 甲子. For added clarity, the system was combined with a regnal year number. The first year was yuán'nián 元年 "first year", then èryuán 二元 "second [year] of the epoch", sanyuán 三元 "third [year] of the epoch", etc, until it reached the sixth year, when it started again. Note that the regnal years use decimals. Tiring of this, Han dynasty (206 BCE – 220 CE) emperors started also using an era name (niánhào 年號) to specifically designate the ruler. The era name was less systematic than the sexagenary cycle, and was more like a little auspicious slogan. Until the Ming dynasty (1368–1644), emperors would change their era names whenever they wanted. From the Ming onwards, emperors tended to stick with one era name throughout their reign.
Traditionally, Chinese count years from one, not from zero, so a person is born one year old (technically one suì 歲 old, as lunar years are not exactly the same as Western years). The same is true for regnal years: the year that a ruler becomes emperor counts as year one. It would also always begin on Chinese New Year's Day.
This system lasted until 1911, when the Xinhai Revolution toppled the last emperor and ended the dynastic system. Xīnhài 辛亥 refers to the year 1911 in the sexagenary cycle. The full year would be Qīng xuāntǒng sānnián xīnhài 清宣統三年辛亥, that is, the xinhai year, the third year since the adoption of the era name xuantong by the Qing emperor Pǔyí (溥儀).
After 1911, they started from year zero and dropped the sexagenary cycle. So 1916 was called the fifth year of the republic (mínguó wǔnián 民國五年). This is still how the years are officially reckoned in Taiwan. After 1949, the PRC adopted the Western system.
Months were less complicated. From the Han onwards they just counted with decimals, although for more serious situations like horoscopes they still used the sexagenary cycle. There are some exceptions though, for example zhèngyuè 正月 is a non-decimal name for the first month of the year, and was quite commonly used.
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