The movie portrays the rise of three African-American women within the ranks of NASA around the time of the first space launches (roughly 1959 through 1962). The character names - all real women - were Katherine Goble (played by Taraji P. Henson), Mary Jackson (played by Janelle Monáe), and Dorothy Vaughan (played by Octavia Spencer). Blessed with superb technical minds, they struggle with institutionalized racism and sexism. Vaughan is working as a supervisor: the supervisor role has been vacant for months and she has the skills but isn't paid for it. Goble and Jackson are "computers" (people who do complex mathematical calculations), with Goble being the most skilled in the building and Jackson clearly both skilled enough and intelligent enough to be an engineer - a position she wants but which clearly couldn't be done by someone with the double disability of dark skin and female gender. Happily, she doesn't accept this valuation. All three work hard and fight for what they want.
The casual racism and sexism is horrifying when viewed from Canada in 2017, but I guess that was the point. The characters are well drawn and well acted, and it's a really interesting window into a piece of American history - as odd as the juxtaposition of science history and race relations history may seem. This particular aspect of it finds three very intelligent women struggling for the right to be allowed to be the best, and to be acknowledged for it.
There's some slight irony that the worst of their fight seems to have been over by the time the space launches of the movie happened: the Historical accuracy section of the Wikipedia entry about the film is definitely worth a read, although preferably after you've watched the movie. I find it an acceptable compromise to show their struggles and achievements in parallel with the great moments of technical achievement they were involved in - all wrapped in a two hour movie. If anything, after reading that section in Wikipedia I felt the movie may have somewhat understated their achievements ... Definitely worth a watch.