In "The Gift of the Magi," the narrator tells us that the young couple's apartment had "a letter-box into which no letter would go and an electric...

As the story opens, we see Della counting her meager pile of coins, realizing that it's almost Christmas, and flopping onto the couch to cry. She's in her apartment, which she rents (she doesn't own it) and which includes furniture in the rent (meaning she doesn't own her furniture either).

While that's going on, the narrator describes the apartment for us, saying: 

"In the vestibule below was a letter-box into which no letter would go,...

As the story opens, we see Della counting her meager pile of coins, realizing that it's almost Christmas, and flopping onto the couch to cry. She's in her apartment, which she rents (she doesn't own it) and which includes furniture in the rent (meaning she doesn't own her furniture either).

While that's going on, the narrator describes the apartment for us, saying: 

"In the vestibule below was a letter-box into which no letter would go, and an electric button from which no mortal finger could coax a ring."

We already know that Della must be very poor, and now we find out that the letter-box is pretty much broken, and so is the doorbell. It sounds like maybe the lid of the letter-box is stuck, so you can't shove a letter into it; also, perhaps you can push on the button, but it won't make the ringing sound that it's supposed to. 

That makes sense: Della's in such poverty, so she probably can't afford to fix those things. She's not thinking about them, anyway; she's focused on how she can scrape up a little more money to get a Christmas present for her beloved husband.