The Office of the First Lady of the United States (FLOTUS) is the official host of the White House and is traditionally the President's Wife, but several presidents have had non-spousal First Ladies including the President's Mother Sister, or Daughter (in these cases it was not incestuous marriage either, just so we're very clear on that) in cases of widower Presidents or Grover Cleveland, who never married.
Typically, the First Lady's official duties are to greet guests and arrange any social functions or gatherings at the White House (her office will be in charge of making the arrangments for a State dinner for example or this year's Easter Egg Roll) as well as the decorations during holidays and the living space of the First Family. Though I'm not sure, the office may also run errands for the First Family that the First Lady couldn't do (It's rather hard to have an in and out pick up of milk at the grocery store when you have a Secret Service Detail).
While it's more notable today, the First Lady has always been an important figure in political society in Washington D.C. since the days of the City's namesake, and the First Lady is typically seen a sort of style and fashion icon by the nation, and will frequently be asked for thoughts on many topics in the realm of the nations culture so the office also coordinates the press around her and frees up the West Wing press office to focus on issues relating to the running of the country. Typically the First Lady's budget isn't discussed as something to be cut unless budget allocated for the White House writ large is being discussed... even then, it's the President's office that gets the closer look as they tend to be larger and more numerous areas to cut. In addition, in American Politics, there is a taboo about targeting the character of a politician's family (adult family members are fair game if they first make the commentary. Attacking the President's still minor Chidren is considered uncalled for. Attacking a President's Dog is treated with a level of scorn reserved for people who deserve the death penalty. No less than three Presidential pooches became famous and beloved after someone crossed that line (FDR's Dog Fala, Nixon's Dog Checkers, and George H.W. Bush's service dog who famously stood guard over his master's casket during the course of a viewing)). So the matter of funding the office of FLOTUS is typically not seriously discussed as it tends to fall too close to the taboo for even the most ardent budget cutter to survive making it the sole cut.
Edit: Though it never happened, the appropriate title for the male spouse of a President is "First Gentleman" and has been used for the husband of Female Governors of States in the United States as well as the English title for a Female President's husband in other nations with a President (unless that nation uses some other title for the role). There were jokes about what someone would call the male spouse in 2016, as Bill Clinton would potentially have the role, with a joke about addressing him as First Lady Bill Clinton that were more commentary on the character of Bill than serious proposals to keep the title for both genders. As I said earlier, FLOTUS is given to the female occupant of the office, regardless of her status as the President's Wife.