Barbara, like Susan, was one of the first companions of the Doctor. Barbara was Susan's history teacher at the Coal Hill School in London. She grew suspicious of Susan's intelligence - low expectations, I guess - and recruited her fellow teacher Ian Chesterton to join her in following Susan home.
Here's Barbara and Ian studying Susan, who is not acting suspiciously in the slightest.
Finding Susan's home to be a Police Public Call Box in the middle of a junkyard, Barbara's suspicions were not alleviated. She pushed her way into the TARDIS, and the Doctor, very old and crotchety then (as opposed to young and hot, like today), was pissed that he and Susan's cover had been blown. He set the TARDIS in flight, absconding with Barbara and Ian. At this period in the show's history, the Doctor had no control over the TARDIS whatsoever, so the chances of the two schoolteachers returning to their own time were slim to none.
Here they all are in the original TARDIS set, in the first story, An Unearthly Child. Doesn't Barbara look amazing? She's all ready to wallop a Dalek with her gigantic handbag.
Speaking of which, Barbara was the first ever Doctor Who character to encounter a Dalek.
And here's that historic moment, from the second serial, The Daleks. Although understandably alarmed here, most of the screamy bits were left to Susan. Barbara was tough, brave, witty, and smart. She stood up to the Doctor when he was being a jerk, which happened quite often in the early days.
Barbara and Ian stayed on with the Doctor after he ditched Susan. In the next story, The Rescue, they met up with Vicki, an orphan from the future. Barbara got on fairly well with Vicki, despite killing her pet sand beast with a ray gun when they first met, mistaking it for a threat.
Here she is with Susan in the fourth serial, Marco Polo. I can go on and on about how awesome Barbara was, but without pictures you can never truly appreciate the magnificence of her hair. It was the fourth companion, really.
When the TARDIS landed in fifteenth century Mexico, the Aztecs mistook Barbara for a goddess. Here she is, resplendent in feathers. Barbara was all, "The Aztecs were awesome, except for the human sacrifice thing, so I'm going to change that." And the Doctor was all, "You can't rewrite history, not one line." And she was all, "Screw you, I'm doing it anyway." It didn't end well. When you're considering changing history, bet on the Time Lord over the history teacher.
Again, please note the hair. There must be a great salon in the TARDIS.
After an occasionally thrilling chase through time and space by the Daleks, Barbara and Ian used their abandoned time machine to finally get back home. Barbara's fate after leaving the TARDIS would go unrevealed until a recent episode of spin-off The Sarah Jane Adventures, which suggests she and Ian became professors at Cambridge and haven't aged since the '60s.
If you couldn't tell, Barbara is one my favorite companions. Although the '60s Doctor Who girls weren't all quite as useless as is frequently said, it would be a long time before the show saw a female companion quite as capable as her. And it would never again see a companion with hair as awesome.