While the previous answer makes some very good points, it also leaves out important factors.
First, we must understand that the social reform movements (particularly abolitionism) of the 1830s and 1840s helped lead to the women’s suffrage movement. Women became very involved in these movements. As they became more politically involved, they started to question the idea that they were not capable of voting. They particularly questioned this as they fought hard for the rights of African Americans when some of the rights they were demanding for blacks were not available to them. The abolitionist movement is seen as the foundation of the suffrage movement and cannot be ignored in an answer to this question.
Second, we must understand that the Progressive movement had a lot to do with women getting the vote in 1920. Progressives were very much concerned with improving society. They felt that letting women vote would be one way to do this. They wanted, for example, to ban alcohol. That was an idea that had much more support among women than it did among men. In general, there was still the attitude in those days that women were morally better and had more social conscience than men. The Progressives felt that women would be their natural political allies and therefore they helped get women the right to vote.
We cannot understand the women’s suffrage movement without understanding the role of the social reform movements of the antebellum era and that of the Progressive Era.
The factors that lead to the women's suffrage movement and to the eventual adoption of the 19th Amendment to the United States Constitution were relatively straightforward: in effect, women objected to being denied the right to vote in elections and to run for public office. Women were denied that right simply on the basis of gender. They were not seen as intellectually sophisticated and comprised of the gravity required of an informed citizenry.
The women's suffrage movement in this country dates to the pre-Revolutionary War period, when a small number of women began to agitate for the right to vote in elections. The movement slowly grew over the next century, with increasing numbers of women joining the movement. A New York woman named Susan B. Anthony joined the cause in the mid-19th Century, and became one of the more prominent standard-bearers for the suffrage movement.
A small number of states, particularly Utah, adopted the gender-neutral voting laws, but at the federal level, Congress continued to resist. The movement reached its climactic stage during the period of the administration of President Woodrow Wilson, who opposed suffrage. Demonstrations outside the White House in 1917 resulting in the arrest of some of the women followed by a hunger strike by one of the imprisoned demonstrators, Alice Paul, compelled President Wilson to shift his position on the issue. On June 4, 1919, the United States Senate passed an amendment to the United States Constitution that prohibited the federal and state governments from prohibiting someone to vote on the basis of gender. By 1920, enough states passed resolutions of approval for the amendment to become the 19th Amendment to the Constitution.