The governor of Springfield's fair state, whichever state that is.

She's a fairly bland presence in this episode, the focus is on Burns' campaign, so we see very little of Bailey and her policies or personality. As the show explores political satire further in the future, characters like this will be fleshed out to comment on the corruption or ineffectiveness of politicians (like with Quimby, who has not been defined as a monster yet, but will be.) For the purposes of this story, all Bailey needs to be is Not Mr Burns.
But then Mary Bailey comes back in season 14's "Bart vs. Lisa vs. The Third Grade", over a decade later, and I find this quite interesting (I recommend watching her in this scene, Tress MacNeille's repeated cries of agony are very funny.)

By the time we get to season 14, the show starts revisiting some of the lore it has accidentally built up over the 14 years it has been on the air - there was no need to reuse the same governor from season 2 here, no one would have really noticed. But by this point in the production, the writers had started recording audio commentaries for the early seasons - revisiting early episodes that they hadn't seen in a while. I believe that in rewatching early episodes for the DVDs, they were inspired to revisit some concepts from them. Mary Bailey is a good example of this - the season 2 DVD was released in 2002, as was season 14. The concepts for "Bart vs. Lisa vs. The Third Grade" would have been fleshed out around the same time that the "Two Cars In Every Garage And Three Eyes On Every Fish" commentary was recorded, and someone in the production pipeline could have thought "Hey, that governor character, we should use her again." It's not a major revelation, but I find it interesting that as the show goes on, the weight of the world they've built becomes inescapable.
After this, Bailey is firmly established as governor, appearing a few more times. As a result, we are given a glimpse into how ineffective she is as a governor, such as in season 16's "The Seven-Beer Snitch", where she releases a bunch of prisoners because there's no room for them elsewhere.


This is the Simpson way: exist long enough in their world and you'll be a walking joke.
A couple of odds and ends about Mary Bailey: she strikes a strong resemblance to Beatrice Simmons, presumably because this is just now old women were stylised in the show in this era.

And also she shares a name with a main character from "It's A Wonderful Life", played by Donna Reed.

Were I to speculate, I'd say this shared name is probably intentional, but not significant. I suspect the writers will have been very familiar with this classic film and just included it as a meaningless reference, but if you see any similarities between the two characters, feel free to let me know! Rampant speculation is half of the fun here.
This post is part of my "Every Simpsons Character Ever" series. For a list of my rules in this project, click here.