Just like her husband, she does not possess any magical abilities whatsoever. It is unknown what her occupation is and until then, it can be assumed she stays home and takes care of the house. She is often seen waking the kids up for school, asking the kids how they're doing at evening family dinners, attending different events or parent-teacher gatherings.
She is friendly and easy-going but stern and demanding when needed, such as when denying Ray Ray different bank cards or when expecting June to go to school despite June faking being sick. She sees through most of her kids' ruses and moods and thus could probably find out about June's double life fairly easily if it weren't for Monroe, their family's dog, freezing her or dusting her memory every once in a while when she's too close to truth. Ray Ray is appalled at Monroe not having told him this sooner, as it could have helped him out a few pinches with his mother.
In Party Monsters, she asks Michael why his family "always eats like they've been starving on an island" so it can be surmised that she has a snark level as great as her kids and possibly greater than that of her husband's. These snide remarks are usually said under the breath while the other character is not paying attention or is too oblivious but she simply says it out to her family members freely.
She has been seen being friendly with Ophelia's mom so it can be concluded that the two could possibly be friends either through their daughters' friendship or simply from knowing each other from their daughters' school. It could also be that they live close by. She also seems to be good friends with Roger's mom but this might be more due to both their husbands sharing a work place. In Feets Too Big where they share an episode, she sees Mrs. Radcliffe make a large dinner out of a travel pack with zero effort and immediately remarks Dennis to not even look that way when he notices it.
Not in My Backyard
I'll Get By with a Little Help from My Elf
Take My Life, Please
O, Brother, What Art Thou?
The Great Escape
Meet the Parent
There's No Mitzvah Like Snow Mitzvah
Adventures in Babysitting
Feets Too Big
Water We Doing Here?
Adventures in Babysitting
Her first name is most likely a reference to DC Comic's Batgirl A.K.A. Barbara Gordon; creator Judd Winick being a comic book writer and fan, the show is littered with comic book allusions.
Barbara and her husband seem to enjoy going out to do activities together, such as seeing musicals on ice or going camping in the woods.