Early years
Using a telescope, she discovered "Miss Mitchell's Comet" (Comet 1847 VI, modern designation is C/1847 T1) in the autumn of 1847. Some years previously, King Frederick VI of Denmark had established gold medal prizes to each discoverer of a "telescopic comet" (too faint to be seen with the naked eye). The prize was to be awarded to the "first discoverer" of each such comet (note that comets are often independently discovered by more than one person). She duly won one of these prizes, and this gave her worldwide fame, since the only previous woman to discover a comet had been Caroline Herschel.
There was a temporary question of priority because Francesco de Vico had independently discovered the same comet two days later, but had reported it first; however, this was resolved in Mitchell's favor. The prize was awarded in 1848 by the new king Frederick VII.
Note: some biographical sources misinterpret the words "first discoverer of a telescopic comet" and report that she was the first person in history to discover a comet using a telescope. That is not the case.
She was the first professional woman astronomer in the United States, noted for her discovery that sunspots are whirling vertical cavities and not, as previously thought, clouds.

Comet discovery
She became the first woman member of the American Academy of Arts and Sciences in 1848 and of the American Association for the Advancement of Science in 1850. She later worked at the U.S. Nautical Almanac Office, calculating tables of positions of Venus, and traveled in Europe with Nathaniel Hawthorne and his family.
She became professor of astronomy at Vassar College in 1865, the first person (male or female) appointed to the faculty. She was also named as Director of the Vassar College Observatory. After teaching there for some time, she learned that despite her reputation and experience, her salary was less than that of many younger male professors. She insisted on a salary increase, and got it.

Career
In 1842, she left the Quaker faith and followed Unitarian principles. In protest against slavery, she stopped wearing clothes made of cotton. She was friends with various suffragists such as Elizabeth Cady Stanton and co-founded the American Association for the Advancement of Women.

Efforts
She died in June 28, 1889, at the age of 71, in Lynn, Massachusetts. She was buried in Lot 411, Prospect Hill Cemetery, Nantucket.

Online sources

Kendall, Phebe Mitchell. Maria Mitchell; Life, Letters and Journals. Boston: Lee & Shepard, 1896. (out of print; written by her niece)
M. W. Whitney, In Memoriam, (Poughkeepsie, N. Y., 1889)
M. K. Babbitt, Maria Mitchell as her students Knew her, (Poughkeepsie, N. Y., 1912)
Albers, Henry editor "Maria Mitchell, A Life in Journals and Letters" College Avenue Press, Clinton Corners, NY, 2001. (Henry Albers was the Fifth Maria Mitchell Professor of Astronomy at Vassar College.)