the 1900s
Refers to the decade from 1900 to 1909, or sometimes loosely to the entire 20th century. Context usually clarifies the intended range.
In the early 1900s, industry developed rapidly in many countries.
This photo captures the fashion of the 1900s well.
千九百年代 specifically refers to the decade 1900–1909 (or loosely the 20th century), while 二十世紀 always means the full 100-year span from 1901 to 2000.
Compound of 千 (thousand), 九 (nine), 百 (hundred), 年 (year), and 代 (era/decade). Literally 'the thousand-nine-hundred-years era'.