A century is one hundred of something, usually one hundred consecutive years, or 100 runs in cricket, or a bicycle ride of 100 miles in a day.
In all dating systems, centuries are numbered ordinally. Thus, one speaks of the First Century of the Common Era, never of Century Zero.
There is considerable disagreement about whether to count the centennial year (i.e. 2000) as the first or last year of a century. This confusion is documented for every centennial year from 1500 onward, and almost certainly arises from the introduction of Arabic numerals and the concept of zero to Western Europe in the twelfth century.
The oldest dating systems were regnal, and considered the date as an ordinal, not a cardinal number. Thus, one speaks of the first year of the reign of King so-and-so. Obviously, the century problem does not arise in such systems. Somewhat later, systems arose dating from the founding of a dynasty, city or religion, and these continued ordinal, rather than cardinal, counting. Thus Ab Urbe Condita counts the Year 1 as the founding of Rome; Anno Domini as the first full year of Jesus's life; the Islamic Calendar as the year of the Hejira, so it is often latinized as Anno Hejira.
Under any such ordinal system, the first century comprises the first through one-hundredth years, the second century the hundred-and-first through two-hundredth and so on. When roman numerals were used to describe a year, it was still generally read as an ordinal: A.D. MLXVI is the one-thousand and sixty-sixth year of our lord. Once Hindu-Arabic numerals came into common use, the number tended to be regarded as a cardinal number. When the concept of zero became widely known, it became almost irresistible to regard 2000 as the first year of a new century, and most people so celebrate.
More modern systems of dating, (such as the astronomical calendar, see proleptic Gregorian calendar) begin with a year zero. In these cardinal dating systems, it is perfectly logical to use 0 to 99 as the first century, and to regard 2000 as the first year of the twenty-first century.
This encyclopedia has a page for each century; see Centuries.
"The Battle of the Centuries", Ruth Freitag, U.S. Government Printing Office. Available from the Superintendent of Documents, PO Box 371954, Pittsburgh, PA 15250- 7954. Cite stock no. 030-001-00153-9.