It's all very methodical, with the book divided into three parts. The first part has Poirot asked to investigate a murder. There's nothing unusual about this in of itself--but this particular murder was committed 16 years earlier. A well-known artist had been poisoned. His wife was convicted of the crime and then soon died in prison. The evidence pointing to her guilt seems conclusive.
But now the couple's daughter--a 5-year-old at the time of the crime, but now an adult, wants Poirot to look into the case. Based on a letter written to her just before her mom died, the daughter is convinced of her mother's innocence.
The first part of the novel concludes as Poirot visits each of the suspects in turn. He thinks of them as the "five little pigs," metaphorically assigning each of them an equivalent to one of the pigs in the nursery rhyme.
Poirot gets each of them to write an account of the day the murder happened, though he appreciates that so much time has elapsed that there will inevitably be gaps in their statements. This is Part 2 of the novel.
Of course, another important reason she's still popular today is her ability to create magnificent fair play puzzles. We hardly ever figure out whodunnit before Poirot or Miss Marple does, but when its all explained to us, we always have to admit the clues were there for us to see.
In this case, some of the clues involve understanding the psychology of those involved in the case, but conclusions made about that are all reasonable and do nothing to distract from the "fair play" aspect of the story.
The last part of the novel is Poiret visiting each of the suspects in turn, asking each of them one additional question. Then he brings them all together to explain what REALLY happened on that fateful day. The fate of the real killer is an interesting and atypical one, but dramatically satisfying.
Whenever I read a great murder mystery, it always brings back my desire to one day use brilliant deductive reasoning to solve a murder, then do a melodramatic summation with all the suspects listening. But, as I've written before, I have sadly boring friends who regularly fail to get murdered or (at least) get falsely accused of murder so I can save them by finding the real killer. This is a source of unending disappointment for me.