What does the narrator mean by describing the couple's apartment in "The Gift of the Magi" as having "a letter-box into which no letter would go and an electric button from which no mortal finger could coax a ring"?

The narrator means that the couple in the story, Jim and Della, live in near squalor. They pay $8 a week for a "furnished flat," meaning an apartment that comes with furniture (this arrangement was common at the time, as furniture was relatively very expensive for people). Their letter box in the hallway is too small to fit letters, and their buzzer doesn't work. They have a pier glass, which is a mirror on the wall between two windows, and they have to glimpse themselves in it by moving between the strips of glass to see their reflection (an art that Della has mastered with practice). They live simply and have only two luxuries--Della's beautiful brown hair and Jim's gold watch--and they have very little money for extras, such as Christmas presents. It's for this reason that Della sells her hair and Jim sells his watch to provide a Christmas present for the other person. 

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.

What does this quote from "The Gift of the Magi" mean: "...a letter-box into which no letter would go and an electric button from which no mortal finger could coax a ring."

A lot of the rooming houses and apartment houses in Manhattan were old and not well maintained. They had originally been private residences for people with big families. Typically they were three and four stories tall. The servants lived on the top floor. The original owners' children grew up and left home. The parents did not want to continue living in such big houses. In some cases one of the parents would be dead, and the survivor was all alone in a big decaying building. It was nearly impossible to sell such buildings to single families, so they were converted to rooming houses and apartment buildings. Even the basements were divided up with partitions and converted to rental units. The buyers of such buildings did not maintain them well because the land was worth more than the buildings, and they knew that eventually they would tear the old structures down and replace them with modern high-rise buildings, either commercial or residential. This sort of thing happens in all big cities in America. Real estate goes through its own process of evolution. The letter-box cannot be opened to receive mail, probably because the key was lost a long time ago. So the postman probably dumps all the tenants' mail in one pile near the boxes and lets the tenants find their own. And the electric button, which would ring a bell in the Youngs' little apartment if it worked, went out of order and was never fixed. If Jim or Della complained to the owner of the building they might have their rent increased. The landlord might be the XYZ Corporation and own a hundred such buildings. These two features--the letter-box and the electric button--are sure signs that the building is doomed to eventual demolition. As O. Henry says in his story:

A furnished flat at $8 per week. It did not exactly beggar description, but it certainly had that word on the lookout for the mendicancy squad.

O. Henry gives a more detailed description of such a building in his story "The Furnished Room." Here is a sample of his marvelous description from that story:

The young man followed her up the stairs. A faint light from no particular source mitigated the shadows of the halls. They trod noiselessly upon a stair carpet that its own loom would have forsworn. It seemed to have become vegetable; to have degenerated in that rank, sunless air to lush lichen or spreading moss that grew in patches to the staircase and was viscid under the foot like organic matter. At each turn of the stairs were vacant niches in the wall. Perhaps plants had once been set within them. If so they had died in that foul and tainted air. It may be that statues of the saints had stood there, but it was not difficult to conceive that imps and devils had dragged them forth in the darkness and down to the unholy depths of some furnished pit below.

O. Henry must have lived in such places during his years in New York. You get what you pay for. People flock to New York from all over the world, and they drive up the rents on the relatively small island of Manhattan in accordance with the iron law of supply and demand. Another decrepit building which had seen better days is the setting for another O. Henry classic story "The Last Leaf."