These days, if you asked someone who the first female Director of S.H.I.E.L.D. was, they'd most likely answer "Peggy Carter" or "Maria Hill." To be fair, those aren't completely unfounded answers. Peggy was the first to hold the title in the Marvel Cinematic Universe and Maria Hill has held the position more recently. However, in Marvel comics, the first female Executive Director of S.H.I.E.L.D. was actually Sharon Carter.
The timeline of Directors goes like this (minus repeat stints): Rick Stoner, Nick Fury, Timothy "Dum Dum" Dugan, G.W. Bridge, Sharon Carter, Maria Hill, Tony Stark, Norman Osborn, Steve Rogers and Daisy Johnson. Here's how Sharon took on the role -- and how Marvel seemed to forget it.
DEATH AND REBIRTH
Back in 1979, prominent S.H.I.E.L.D. agent Sharon Carter was seemingly killed in the line of duty. Captain America #233 tells the story of how she investigated and infiltrated the National Force, a white supremacist terrorist group. While taking down the organization, she was subjected to their brainwashing and a videotape made it look like she killed herself.
In 1995, Mark Waid revived her in Captain America #444. Waid revealed that Sharon's death was faked so she could go on a top-secret mission for S.H.I.E.L.D. However, the mission didn't go as planned and Nick Fury, the Executive Director at the time, left her behind enemy lines, believing her to be dead.
Waid told Wizard Magazine in 1997 that he decided to revive Sharon because he loved the feisty nature of her character and how she played off of Steve. "The reason she works so well with Cap is because she's a complete cynic and he's a complete idealist," Waid commented.
While behind enemy lines, Sharon was subjected to all kinds of torture. She was starved, caged and sexually assaulted. The horrors she faced during this time understandably hardened her and led her to hold a grudge against Nick Fury after she returned. She denied any and all requests to rejoin S.H.I.E.L.D., operating as a freelance spy instead.
REJOINING S.H.I.E.L.D. AND BECOMING DIRECTOR 
After the Fury/Agent 13 miniseries, Sharon finally made peace with Nick, saving him from a pocket dimension. Then, in Dan Jurgens' Captain America #31, Fury takes a leave of absence from S.H.I.E.L.D. Directorship to investigate the disappearance of agent David Ferrari.
At this point in time, Sharon and Steve Rogers, aka Captain America, were heavily on-again/off-again. The two clearly loved one another, but Sharon refused to start their relationship again because she was afraid of getting hurt once more. During one particularly dramatic event, the two almost rekindled their relationship. But Sharon ultimately shot him down because she had to focus on her career now that she was Director of S.H.I.E.L.D.
Sharon remained Director of S.H.I.E.L.D. for 14 issues, published from July 2000 to September 2001. Because of her close relationship with Steve, the two worked well together, despite engaging in the occasional bickering they're known for.
She interacted with the Avengers, the Thunderbolts, the X-Men and dealt with threats like The Hand trying to turn downtown Tokyo into K'un Lun. She was a natural fit for the job, as noted by Dugan and Captain America, who praised her leadership and capabilities as a field agent.
Sharon was always known for being a no-nonsense type of worker who put her job first. As Director, she commanded respect. There were times where Nick Fury and other top S.H.I.E.L.D. personnel even had to answer to her.
However, Sharon's tenure as Director was always temporary and in issue #45 of Dan Jurgens' Captain America run, Nick Fury returned from his leave and took back his position... without telling her.
Needless to say, Sharon wasn't too happy.
She marches up to him, saying, "I did a fine job as director, Nick! Why didn't you tell me you were taking the chair again?" Nick says that his business with David Ferrari is done and he needs Sharon for another mission. "I'm kicking you over to a more important assignment 'cuz yer the best field agent I got!" he exclaims, putting her in charge of taking down the Red Skull with Steve.
MARVEL...FORGOT?
Sharon's tenure as Director was relatively short, but it brought her back into the S.H.I.E.L.D. fold. Afterward, she retained her position as Agent 13, the best field agent S.H.I.E.L.D. had to offer. In Ed Brubaker's Captain America run, she becomes the liaison to Captain America, running missions with her sometimes-boyfriend. In Captain America: Steve Rogers, she became Commander Carter, climbing the ranks of the government agency once more.
However, Marvel seems to have forgotten this bit of canon (or chosen to ignore it). In Captain America: Steve Rogers #9, the S.H.I.E.L.D. Director at the time, Maria Hill, was on trial. The Tribunal of the World Security Council convened to review Hill's performance, determining whether or not she was fit to serve the role after a series of mishaps.
At this time, unbeknownst to the rest of the Marvel Universe, "Steve" was really a Hydra Agent created by Kobik as part of writer Nick Spencer's controversial Secret Empire storyline. Rogers is called to the stand, declaring that Hill is unfit to remain as leader and suggests Sharon Carter take the position instead. Hill defends herself, claiming that Sharon would have no idea how to be Director.
A statement that is obviously false considering Sharon was Director before Maria Hill was even a character in the Marvel Universe. In the MCU incarnation, Sharon's role as the first female director of S.H.I.E.L.D. was given to her great aunt, Peggy Carter. Since then (and with the help of other MCU-inspired comics and portrayals), Peggy has become more synonymous with the title of "S.H.I.E.L.D. Director."
Who knows, maybe some day Sharon Carter will get another stint as Director of S.H.I.E.L.D. -- the job is a revolving door after all.