The history of feminism is generally categorized into four waves. The ones mentioned in this question are the first wave (roughly taking place from the 1840s through the 1920s) and the second wave (roughly taking place between the 1960s and 1980s). Both of these waves focused on combating the ways women were limited in society. However, the issues they chose to focus on were very different.
First-wave feminists were mostly interested in suffrage and property rights. Women were officially regarded as second-class citizens, unable to vote or own property in the same way men could. Women such as Susan B. Anthony, Ida B. Wells, and Elizabeth Cady Stanton campaigned for these rights. Conventions for women's rights started gaining steam around the 1850s, and by the 1920s, the dream of these suffragettes would be fulfilled with the ratification of the Nineteenth Amendment.
Second-wave feminism generally fought the unofficial ways in which women were limited in society. Though women could vote and own property, they were not paid the same wages as men in the same professions, birth control was still quite controversial, and women were usually expected to marry and have children. Second-wave feminists sought to liberate women from roles that they as individuals might not find fulfilling.