The Earthly Branches (or Chinese Zodiac) are the animals that represent each of a 12 year cycle. These are paired with 10 Celestial Stems to form a 60 year cycle.
Earthly Branches:
Celestial Stems:
Thus, the cycle begins with jia-zi, yi-chou, bing-yin, etc. ending with gui-hai.
Although the initial yearly cycle, as explained on the main page, was based on an astronomical event, subsequent numbering of the years has been restarted at the beginning of each emperor's dynasty (or new civil government), or at any time within a dynasty that the emperor declares.
The months are simply numbered from 1 to 12, each beginning with the new moon (thus some have 29 days, others 30, but it isn't fixed), and intercalculating a thirteenth month in leap-years to keep the calendar in line with the solar year. This extra month is given the same number as the previous month, but is designated as intercalary. Thus a normal year has from 353 to 355 days, and a leap-year has from 383 to 385 days