Victoria Claflin Woodhull (1838-1927) was the first woman to run for president. She ran under the banner of the Equal Rights party, not a major political party of the time. Since she ran about 50 years before the 19th amendment was ratified, giving women the right to vote, Woodhull could not even vote for herself. She lost, winning zero electoral votes. It would be close to a century later when another woman would run for her party's nomination for president. (Maine Senator Margaret Chase Smith ran in 1964)
So who does the first woman pick as her running mate? The Equal Rights Party named former slave and abolitionist Frederick Douglass as her running mate. But Douglass never actually accepted the nomination and actually gave stump speeches for Woodhull's competition, incumbent President Ulysses S. Grant.