The first century consisted of the years 1 through 100. Therefore, the second century consisted of 101 through 200, the third century 201 through 300, and so on. That’s why the 19th century, for example, consists of the 1800s instead of the 1900s.
Centuries aren’t normally capitalized. Some publications spell out centuries (e.g., twentieth century), while others use numerals (e.g., 20th century) for centuries after the tenth. It’s a matter of preference, so neither is inherently right or wrong.
Centuries and hyphens
It has 22 bedrooms, and joined to the priory on the first floor is the 17th-century Gothic-style Chapel of St John. [Guardian]
Nineteenth-century French novels have a lot to say about our 21st-century world. [Huffington Post]
As a noun phrase, a century has no hyphen—for example:
The last time an enemy army threatened American territory was in the early nineteenth century … [Canada Free Press]
We, the theatergoers of the 21st century, are posterity itself … [NY Times]