Maria is one of the strangest characters in Dubliners because, despite the fact that she's one of the main characters of a story, she doesn't want to escape her life, doesn't seem particularly unhappy, and is by no means self-destructive in her behavior. That does not mean, however, that her story isn't a difficult tale of paralysis and sadness. It's just that we have to read between the lines to get a hold of it.
Maria used to be a nanny for a family with two boys, Joe and Alphy. She stayed on after the boys grew up, at least until a big fight separated the family and kept the boys apart from each other. When that happened, the boys found a job for Maria, and a place for her to stay, and while Maria seems okay with it, it's not entirely classy. The Dublin by Lamplight Laundromat is a place for homeless and otherwise un-attached women to work in a laundry and get meals and a place to stay. Maria isn't one of these women; she just helps them out and works in the kitchen. But still, it's not glamorous, and it did require Maria, a Catholic woman, to spend a lot of time with Protestants.
It's important, by the way, that her name is Maria, especially because that was the last word of the previous story in Dubliners, "Counterparts." There, a young boy who was being beaten screamed that he would say a Hail Mary for his father if he stopped beating him. It's a chilling scene, and it sort of shows how Mary—and the Catholic faith in general—is the way that many people try to relieve their suffering and don't get very far in doing it.
In "Clay," Maria often plays the role of mediator—the person who solves conflicts. Her boss calls her "a veritable peace-maker," and this goes along perfectly with her name, because one of the lines in the Hail Mary is "pray for us now and at the hour of our death." The belief is that the biblical Mary, Mother of Jesus, (that's "Maria" in Latin, by the way) is an intercessor. That means that she's someone who "stands between" humans and God, and pleads for God to do something about the individual concerns of humans.
As it turns out, Maria "stands between" in more ways than one. She really wants Joe to reconcile with his brother, Alphy, for one thing. So she decides to "put in a good word for Alphy" when she sees Joe. This doesn't go too well: "Joe said that Alphy was no brother of his" (Clay.18). And she's also worried about standing between Joe and his wife, and being a burden, and that's why she doesn't go to live with them.
Finally, Maria stands between Joe and the past, and connects him to it. When he asks her to sing, she chooses a song about a dream of wealth and desire: "I dreamt that I dwelt in marble halls" and "that you loved me still the same" (Clay.21). On the one hand, this must make Joe remember that his past, with Maria as his nanny, was like living in "marble halls" compared to how things are now. He even says, "there was no time like the long ago" (Clay.22). On the other hand, it has to make Joe realize that the way the past shows up in the present—with Maria forgetting the verse of a song and losing her plumcake and traveling a long way to visit him—is not really the way he hoped things would be.
In a lot of ways, then, Maria is a sore reminder that life doesn't always turn out how you want it. It may be especially painful for Joe to recognize this because it really doesn't seem to bother Maria. She's the kind of person that it's easy to pity because she's not really self-aware enough to realize that she's single, working in a shelter, and destined to be separated from the people she loves most. But on the other hand, who are we to judge?
The gentleman makes room for Maria to sit down on the Drumcondra tram, the one that takes her to Joe's house. She's surprised that he's so polite, especially compared with the young men on the tram: "He was very nice with her, and when she was getting out […] she thanked him and bowed, and he bowed to her and raised his hat and smiled agreeably" (Clay.13). Even though he's a little bit tipsy, she doesn't reject him: "how easy it was to know a gentleman even when he has a drop taken." When Maria can't find her bag of plumcake when she arrives at Joe's, however, one possibility is that the elderly gentleman has stolen it.
The matron runs the Dublin by Lamplight Laundromat, and always compliments Maria on her ability to solve conflicts between other workers. She also allows Maria to leave work early to make her visit to Joe's house.
Ginger Mooney works with Maria at the Laundromat, and is very "fond" of her. She even proposes a toast to Maria before she leaves to visit Joe's. It's funny to Maria because "she knew that Mooney meant well, though, of course, she had the notions of a common woman" (Clay.8).
Joe is grown-up now, but knew Maria when she was his nanny as a boy. Maria comes to his house to celebrate Halloween with his wife and children. We know that Joe is, even more than everyone else, "fond of Maria." When he was a little boy, he used to say, "Mamma is mamma but Maria is my proper mother," and ever since he has watched out for her. That includes finding her a job at the Laundromat and repeatedly inviting her to come live with his family. The two things that worry Maria are his drinking problem and the fact that he is no longer speaking to his brother, Alphy. The story closes when Joe asks Maria to sing a song, and he gets very emotional hearing her sing a song about being wealthy and desired by men. It seems likely that Joe feels both nostalgia for the past, when his life was simpler and included Maria, and also guilt that she has such a tough life now.