The original Bat-Girl was Betty Kane (first appearance: Batman #139, 1961), the niece of Kathy Kane, Batwoman.
Batwoman and Bat-Girl were created to be romantic interests for Batman and Robin, respectively, as well as wannabe crime-fighting associates. Bat-Girl wore a red-and-green costume to "flatter" Robin. Bat-Girl appeared seven times between 1961 and 1964, but then disappeared in 1964 (along with Batwoman, Ace the Bat-Hound, and Bat-Mite) when new Batman editor Julius Schwartz decided she and other characters were too silly. It has been suggested by scholars that the characters of Batwoman (in 1956) and Bat-Girl (in 1961) were introduced in part to refute allegations of homosexuality in Batman comics; specifically, the enduring claim that Batman and Robin were homosexuals. 
Batwoman and Bat-Girl were later revived in the late 1970s. Bat-Girl even became a member of the Teen Titans West. However, she only appeared four times in this era.