What type of conflict is depicted in "Miss Brill"?

In "Miss Brill" an external character vs. character conflict causes an internal conflict.

Conflict is a struggle.  There are two main types of conflict: internal and external.  An internal conflict is a struggle a character has with his or her self, and an external conflict is a struggle a character has with another.

As in many cases, an external conflict causes an internal conflict in this story.  Miss Brill is enjoying her time in the park, and sees herself as an integral piece of the play.  She is proud of her fur, and appreciates the interactions.

And what they played was warm, sunny, yet there was just a faint chill no, not sadness—a something that made you want to sing.

The external conflict comes from when Miss Brill overhears two young people insulting her.

"But why? Because of that stupid old thing at the end there?" asked the boy. "Why does she come here at all—who wants her? Why doesn't she keep her silly old mug at home?"

She realizes then that these characters do not want her.  This is a character versus character conflict.  The other characters are mean to her, even if they don’t know that she can hear them.

This produces an internal conflict.   Miss Brill feels bad.  She begins to see herself as a burden, and returns to the isolation of her dark and lonely home.

But to-day she passed the baker's by, climbed the stairs, went into the little dark room her room like a cupboard——and sat down on the red eiderdown. She sat there for a long time.

Miss Brill feels lonely and regretful.  She no longer sees herself as part of the great play taking place in the park.  She now sees herself as isolated from the play she used to be a part of.  She imagines she hears crying.  This is a result of the internal conflict she feels.

The story demonstrates how one person’s hurtful words can affect another person so completely as to change that person’s self-perception.  The young couple have no real interactions with Miss Brill.  There is no reason for them not to like her.  Yet they think she is ugly and silly.

What is the conflict in "Miss Brill" by Katherine Mansfield?

Everything was fine with Miss Brill until two people barged in on her psychological life and ruined things for her. As she is sitting in the park, listening to everyone's conversations, imagining herself to be playing a part in the play of life, as Shakespeare says in As You Like It: 

All the world's a stage,
And all the men and women merely players;

two young people enter her space. She incorporates them into the idealistic reverie of her play, telling herself that she, too, has a part, and that if she did not come to the park, people would miss her. They would notice that she was not there.

The young people make fun of her, however. They make fun of the fur collar she is wearing and her looks. They cruelly say they don't know why she even bothers to show up and that she should have just stayed home.

We don't learn what Miss Brill's reaction to the comments are because instead of stopping at the bakery on her way home, as is her custom, she returns to her dark little apartment, unclasps her fur without looking at it, and puts it back in its box - like Eleanor Rigby in the old Beatles' song. She thinks that she hears "something crying." The crying is in her soul because a cruel comment shocks her into the reality that what she thought was happiness might be, instead, loneliness.

The way I see it, there are two conflicts. First, the conflict of "man vs man" - the young couple against Miss Brill because they turn her life upside down in the space of a few moments. After that, Miss Brill has an internal conflict - "man vs himself" because she is now unhappy and "crying on the inside" -- fearful, perhaps, that her life might really not have any meaning.

It is a very sad story in my opinion. It reminds me of the quote, "Where ignorance is bliss, 'tis folly to be wise" (from Thomas Gray). Miss Brill was blissful when she was ignorant of what the young couple thought of her. When she discovers what they really think, she is sad.

See the analysis here on enotes.

What is the climax in "Miss Brill" by Katherine Mansfield?

Unfortunately, you are not allowed to ask multiple questions on enotes, so I have edited your question to focus on the climax of this tremendous short story.

The problem with living life as an illusion is that eventually reality imposes itself and shows the illusion for the untruth that it actually is. This is precisely what occurs in "Miss Brill," as this lonely woman goes to the park on Sunday and distracts herself by watching the various people their and listening in to their conversations. Miss Brill enjoys this weekly sojourn so much because she imagines it to be a kind of massive play, where even she has a part. This allows herself to imagine that she too is important and is significant:

Even she had a part and came every Sunday. No doubt somebody would have noticed if she hadn't been there; she was part of the performance, after all.

She imagines being able to tell the "invalid old man" that she reads the paper to that she is "an actress" and has been one "for a long time." Such rampant escapism allows her to forget the meaninglessness and isolation of her life. Yet, at the climax of the tale, she is forced to accept it. The climax comes when she overhears a boy and a girl talking about her slightingly, destroying her fantasies, and sending her back home in tears:

"No, not now," said the girl. "Not here, I can't."

"But why? Because of that stupid old thing at the end of there?" asked the boy. "Why does she come here at all--who wants her? Why doesn't she keep her silly old mug at home?"

"It's her fu-fur which is so funny," giggled the girl. "It's exactly like a fried whiting."

With these words, Miss Brill is forced to see that this elaborate drama she has created in her mind is an illusion, that her fur that she was so proud of is outdated and she is a lonely woman who is insignificant. This is the moving climax of this story.