A century is one hundred of something, usually one hundred consecutive years, or 100 runs in cricket.
When dating years, centuries are usually taken to begin in the years divisible by one hundred, or alternatively in the year after that, e.g., 2000 or 2001. The use of the odd year came from the Anno Domini system of religious dating, where the year 1 was supposedly the first "year of our lord" and thus the first century would be years 1 to 100 inclusive. If the alternative convention of including a year zero is used (as in the astronomical calendar, see proleptic Gregorian calendar) then it may seem reasonable to use 0 to 99 as the first century.
Wikipedia has a page for each century: Centuries.