There were many reasons women wanted the right to vote, but primarily they wanted equality in all aspects of society and felt their votes would contribute to a more fair and equitable representation of society. The idea of women's suffrage grew from the anti-slavery movement of the early 1800s.
Many abolitionists in the early part of the 19th century were women. The problem, however, was that women had no voice in government. When women tried to voice their opinions at the World's Anti-Slavery Convention in 1840, they were segregated from the men, and their opinions were deemed unfit for consideration.
This led to the Seneca Falls convention in 1848, a women's rights convention from which the idea of women's suffrage was born. The issue of women's right to vote was a controversial topic at the convention, triggering several debates. Women argued that the conditions of society, the state of their communities and religion were of equal concern to them and to men and that they should have equal say in how such matters were governed.
It was not until 1920 that women finally won the constitutional right to vote, though some individual states passed laws allowing women to vote prior to that year.