In the United States during the late nineteenth century, the women's movement was largely concerned with getting women the right to vote. These women were called "suffragettes," and they spent the 1890s, 1900s, and 1910s fighting for the right to vote in elections. Their goal was achieved in 1920, when the nineteenth amendment was finally ratified.
As the other answer mentions, the right to vote was about the only major right for women these female reformers supported that was women-specific. Otherwise, they were usually interested in the temperance movement, which saw alcohol as a major social evil. This related to women's rights in that many women saw alcohol as the cause of much domestic violence. Working conditions in factories and labor reform were also major concerns for women reformers.
The early twentieth century did see some women, such as Margret Sanger, arguing for the right to birth control for a variety of reasons, but this movement is linked with the 1910s, not the late nineteenth century.