(or "The Times, They Are A-Changing!")
When Barbara Gordon makes her "Million-Dollar Debut" as Batgirl, she's portrayed as something of a dilettante crimefighter. Sure, she's bright and capable, but those very same smarts have relegated her to the position of an 'old-maid' librarian with her hair up in a bun. (Actually, it's really up in the same style Princess Leia would later adopt, but that's not the point.) She becomes Batgirl not out of a thirst for justice, but because she wants to get some excitement out of her dull, humdrum life.
But believe it or not, this is actually progress off of the original Batgirl. Betty Kane (niece of Kathy Kane, aka Batwoman) also joins up in crimefighting for the thrill of it; she and her aunt are both portrayed not just as thrillseekers, but as barely competent in their newly-chosen profession. It's a constant struggle for Batman and Robin to keep them out of danger, and the continual subtext of the series is that Kathy and Betty (who use gadgets like lipstick that shoots concealed wires to snare criminals, or mace hairspray) have set "marriage/dating" as a condition of giving up crime-busting. (Which isn't far from the truth--the two characters were introduced solely to prove that Batman and Robin were interested in women, rather than each other.)
The second Batgirl can definitely be seen as progress from the original; she, too, pursues criminals "for kicks", but she's a lot more competent than Betty Kane was. (Although the writers still enjoy tossing in some rampant sexism, such as Gardner Fox's infamous "Batgirl's Costume Cut-Ups" story where Batgirl keeps letting criminals escape while she stops to fix her make-up.) The subtext is no longer, "Women should stop fighting crime and settle down with a good husband," but "Some women feel like there's a void in their life, and they fill it with beating up the forces of evil."
As Barbara Gordon becomes the lead in her own series of back-ups, instead of a supporting character in Batman's series, her role changes further. The writers reflect the changing role of women in the 60s and 70s by having Barbara become more independent, intelligent, and even willing to have a romantic life beyond pining for Robin...and at the same time, not willing to let any of these things take a backseat to her role as Batgirl. Barbara Gordon becomes very much an archetypal feminist as the archetype itself develops, culminating in her going to Washington as a member of the House of Representatives (of the State That Gotham Is In, to paraphrase the Simpsons.)
Alas, right about then is the point where her backups disappear. It's too bad, because "crimefighting Congresswoman" is a great notion for a storytelling engine, far better than "crimefighting librarian." Batgirl's stories tended to have a hard time working her into the action due to her profession--libraries tend not to be the scene of exciting criminal conspiracies. Usually, they preferred to use her relationship with longtime Batman cast member Commissioner Gordon as an excuse to get her involved with a crime. That link was actually one of the best elements of Batgirl's storytelling engine...until the point where there wasn't a Batgirl any more.
Because in the 80s, the whole notion of a "family" of supporting heroes went wildly out of fashion. Kid sidekicks, female counterparts, super-pets, all these became trite and childish in a medium that was desperately trying to be perceived as "adult". Supergirl died, Krypto and Ace the Bat-Hound were written out of continuity, and even Robin had to fight tooth and nail to stick around (with one Robin growing up to take on a more adult, "cooler" persona and another dying.) Certainly, Batgirl wasn't about to buck the trend. So Barbara Gordon's relationship to the Commissioner went from being a useful element of her storytelling engine to an excuse for the Joker to shoot her in the gut without having to reveal her secret identity, and the character went away.
Except that she didn't. Barbara Gordon continued to be a hero and a feminist icon despite being paralyzed from the waist down. Or perhaps because of it--comics have always attracted intelligent people as fans, especially among women, and the idea of a superhero who uses her brains instead of her fists to defeat criminals is one that has deep attraction, especially with the rise of the Internet. Batgirl evolved from being a dilettante librarian to a tech-savvy geek girl, just in time for the Information Age. Her storytelling engine seems to constantly reflect the evolving role of women in society, and her popularity reflects the fact that comics are no longer just a boy's club.