"The 1900's", read aloud as "the nineteen hundreds," means every year that begins with 19-. It can be written with or without an apostrophe: "the 1900's" or "the 1900s."
"The 1900's" means these exact years:
1900, 1901, 1902, 1903, 1904, 1905, 1906, 1907, 1908, 1909
1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918, 1919
1920, 1921, 1922, 1923, 1924, 1925, 1926, 1927, 1928, 1929
1930, 1931, 1932, 1933, 1934, 1935, 1936, 1937, 1938, 1939
1940, 1941, 1942, 1943, 1944, 1945, 1946, 1947, 1948, 1949
1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957, 1958, 1959
1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 1969
1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1978, 1979
1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989
1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999
Thus, "the 1900's" refers to a century.
"The first half of the 1900's" means 1900 through 1949, a total of fifty years. (You could argue about which half 1950 is in. But it doesn't matter. You would never use a vague phrase like "the first half of the 1900's" if you were trying to be precise.)
I don't know a good way to refer to the decade 1900-1909. Sometimes it's called "the oughts" because the British sometimes call zero "ought," and read 1901 as "nineteen ought one." An INDEFINITE period of "1900 and a few years after" can be called "the turn of the century."
"The 1900's" can also be called "the twentieth century" (the 20th century). Yes, this is VERY confusing. Think of it this way: A woman who is 19½ years old is "19," because she has COMPLETED 19 full years. However, she is in the middle of her _twentieth_ year.
Here's a silly technical detail. ALMOST ALWAYS, the "twentieth century" means "the 1900's." But since was no "year 0." So sometimes people say that the first century was the years 1 through 100, and therefore the twentieth century was not 1900 through 1999, but 1901 through 2000.