The film opens with archival footage of police raiding gay bars and arresting patrons during the 1950s and 1960s, followed by Dianne Feinstein's November 27, 1978 announcement to the press that Harvey Milk (Sean Penn) and Mayor George Moscone (Victor Garber) had been assassinated. Milk is seen recording his will throughout the film, nine days (November 18, 1978) before the assassinations. The film then flashes back to New York City in 1970, the eve of Milk's 40th birthday and his first meeting with his much younger lover, Scott Smith (James Franco).
Dissatisfied with his life and in need of a change, Milk and Smith decide to move to San Francisco in the hope of finding larger acceptance of their relationship. They open Castro Camera in the heart of Eureka Valley, a working-class neighborhood in the process of evolving into a predominantly gay neighborhood known as The Castro. Frustrated by the opposition they encounter in the once Irish-Catholic neighborhood, Milk utilizes his background as a businessman to become a gay activist, eventually becoming a mentor for Cleve Jones (Emile Hirsch). Early on, Smith serves as Milk's campaign manager, but he grows frustrated with Milk's devotion to politics, and he leaves him. Milk later meets Jack Lira (Diego Luna), a sweet-natured but unbalanced young man. As with Smith, Lira cannot tolerate Milk's devotion to political activism, and eventually hangs himself. Milk clashes with the local gay "establishment" which he feels to be too cautious and risk-averse.
After two unsuccessful political campaigns in 1973 and 1975 to become a city supervisor and a third in 1976 for the California State Assembly, Milk finally wins a seat on the San Francisco Board of Supervisors in 1977 for District 5. His victory makes him the first openly gay man to be voted into major public office in California and the third openly homosexual politician in the entire US. Milk subsequently meets fellow Supervisor Dan White (Josh Brolin), a Vietnam veteran and former police officer and firefighter. White, who is politically and socially conservative, has a difficult relationship with Milk, and develops a growing resentment for Milk when he opposes projects that White proposes.
Milk and White forge a complex working relationship. Milk is invited to, and attends, the christening of White's first child, and White asks for Milk's assistance in preventing a psychiatric hospital from opening in White's district, possibly in exchange for White's support of Milk's citywide gay rights ordinance. When Milk fails to support White because of the negative effect it will have on troubled youth, White feels betrayed, and ultimately becomes the sole vote against the gay rights ordinance. Milk also launches an effort to defeat Proposition 6, an initiative on the California state ballot in November 1978. Sponsored by John Briggs (Denis O'Hare), a conservative state legislator from Orange County, Proposition 6 seeks to ban gays and lesbians (in addition to anyone who supports them) from working in California's public schools. It is also part of a nationwide conservative movement that starts with the successful campaign headed by Anita Bryant and her organization Save Our Children in Dade County, Florida to repeal a local gay rights ordinance.
On November 7, 1978, after working tirelessly against Proposition 6, Milk and his supporters rejoice in the wake of its defeat. A desperate White favors a supervisor pay raise, but does not get much support, and shortly after supporting the proposition, resigns from the Board. He later changes his mind and asks to be reinstated. Mayor Moscone denies his request, after being lobbied by Milk.
On the morning of November 27, 1978, White enters City Hall through a basement window to conceal a gun from metal detectors. He requests another meeting with Moscone, who rebuffs his request for appointment to his former seat. Enraged, White shoots Moscone in his office and then goes to meet Milk, where he guns him down, with the fatal bullet delivered execution-style. The film suggests that Milk believed that White might be a closeted gay man.
The last scene is a candlelight vigil held by thousands for Milk and Moscone throughout the streets of the city. Pictures of the actual people depicted in the film, and brief summaries of their lives follow.

Question: Which of the several political offices that Harvey Milk ran for did he win?
Answer: Board of Supervisors