What successes did the women's rights movement have in the mid 1800s?

The right to vote "Women's Suffrage".
In 1893 New Zealand was the first self-governing nation in the world where all adult women, including Maori, won the right to vote in national elections

.
Actually, because the Cook Islands (a New Zealand dependancy) held an election earlier than New Zealand, they were the first nation in which aldult women actually voted.
Australia gave the vote to all white women in 1902 (aboriginal women had to wait until 1960!) and the US gave all women nationally the right to vote in 1920.
There are still many countries in which women do not have the right to vote.