Flamebird is the name used by six different fictional comic book characters who have appeared in books published by DC Comics, specifically from the Superman and Batman mythos.
The primary character to use the Flamebird name is Bette Kane, who was the pre-Crisis hero Bat-Girl. However, the original pre-Crisis Flamebird was Jimmy Olsen, who was later succeeded by a Kandorian scientist. In post-Crisis, a Kryptonian hero used the name Flamebird, and in a "One Year Later" storyline, so has Kara Zor-El.
Flamebird characters are also often associated with characters who use the name Nightwing.
Pre-Crisis history
Jimmy Olsen
In pre-Crisis continuity, Flamebird was an alias used by Jimmy Olsen in adventures shared with Superman in the city of Kandor, a Kryptonian city that had been shrunken and preserved in a bottle by the villainous Braniac. After the miniaturized city was rescued from Braniac, Superman placed it in his Fortress of Solitude, where he would occasionally shrink himself, and sometimes one or more guests, to pay a visit to the Kandorian inhabitants.
In Kandor, Superman had no powers and was branded an outlaw due to a misunderstanding. To protect themselves, Superman and Jimmy created vigilante identities inspired by Batman and Robin; however, as neither bats nor robins existed on Krypton, Superman chose the names of two native avian species: Nightwing (for himself) and Flamebird (for Jimmy). At one point, Nightwing and Flamebird teamed up with their inspirations, Batman and Robin, for an adventure in Kandor which would prove especially important to the young Robin.
In Superman #166 (January 1964), the imaginary sons of Superman go to Kandor, and take on the Flamebird/Nightwing personas in order to combat a Kandorian villain by the name of Gann Artar, after finding the costumes used by their father and Jimmy Olsen.
Ak-Var
While in Kandor, Nightwing and Flamebird met Van-Zee, a Kandorian scientist who looked strikingly similar to Superman. At one point, Van-Zee himself donned the Nightwing costume in order to rescue a captured Superman. After Superman and Jimmy's departure from Kandor, Van-Zee took up the role of Nightwing full-time.
Ak-Var, Van-Zee's lab assistant and husband of his niece Thara, later assumed the mantle of Flamebird. The two shared several distinct adventures, once teaming up with Superman and Jimmy.
Post-Crisis
Bette Kane
For a brief time in the 1970s, the young costumed adventurer Betty Kane had joined a west coast version of the Teen Titans, Titans West, under her original moniker of "Bat-Girl". After the Crisis on Infinite Earths, "Bat-Girl" did not exist, though her team did. Secret Origins Annual #3 (1989) established the official post-Crisis history of Titans West. Instead of Betty Kane as Bat-Girl, fans were introduced to a similar character called Mary Elizabeth "Bette" Kane, also known as Flamebird.
The Krypton connection
Nightwing Secret Files #1 tells the post-Crisis tale of how Dick Grayson became Nightwing, but retroactively erases the notion that Superman and Jimmy Olsen ever held the titles of Nightwing or Flamebird, respectively.
The connection between Bette Kane's "Flamebird" and Grayson's "Nightwing" was conjectural until 2001's Superman: The Man of Steel #111, wherein Superman and Lois Lane travel to a version of the Kryptonian past and assume the names themselves. This once again associated Superman with the roles directly, and he revealed to Lois that he had indeed related tales of both Kryptonian legends to Dick and Bette.
Kara Zor-El
In Supergirl #6, Kara Zor-El has assumed the Flamebird identity to fight crime in the city of Kandor, along with Power Girl as Nightwing.