In Agatha Christie's And Then There Were None, the characters die in accordance with a children's nursery rhyme, copies of which are hanging in each of the guest rooms.

The first line of the rhyme is "Ten little Soldier Boys went out to dine; One choked his little self and then there were nine." Anthony Marston is the first character to die. He drinks whiskey that is poisoned with cyanide and chokes to death.

The next line of the rhyme is as follows: "Nine little Soldier Boys sat up very late; One overslept himself and then there were eight." Ethel Rogers is the next character in the novel to die. She goes to sleep on the first night on the island and is found dead the following morning of unknown causes.

The rhyme continues: "Eight little Soldier Boys traveling in Devon; One said he'd stay there and then there were seven." Later on the same day of Mrs. Rogers' death, General John MacArthur dies. He accepts that there is a killer after him and waits by the edge of the water for him/her. The killer hits him in the back of the head.

At this point, the remaining characters realize that the mysterious deaths are consistent with the children's rhyme.

The next line of the rhyme says, "Seven little Soldier Boys chopping up sticks; One chopped himself in halves and then there were six." Thomas Rogers is the next to die. In accordance with the rhyme, he is found dead from an axe wound sustained while chopping wood.

The rhyme continues: "Six little Soldier Boys playing with a hive; A bumblebee stung one and then there were five." Emily Brent dies next. She is injected with potassium cyanide, which is comparable to being stung by a bee, and found dead in the drawing room.

The next line of the rhyme is as follows: "Five little Soldier Boys going in for law; One got in Chancery and then there were four." Justice Wargrave is found dressed as a judge and apparently dead from a gunshot wound to the head. Dr. Edward Armstrong pronounces him dead; however, this is later revealed to be a faked death.

The next line of the rhyme is as follows: "Four little Soldier Boys going out to sea; A red herring swallowed one and then there were three." The novel's next fatality is Dr. Armstrong who is pushed over a cliff and subsequently drowns.

The rhyme continues: "Three little Soldier Boys walking in the zoo; A big bear hugged one and then there were two." William Blore dies next after he is crushed by a heavy, bear-shaped marble clock that is pushed from the ledge of Vera Claythorne's window.

The next line of the rhyme is as follows: "Two little Soldier Boys sitting in the sun; One got frizzled up and then there was one." Vera and Phillip Lombard find Dr. Armstrong's body on the beach. Each believes the other is responsible for his death. Vera suggests they move the body, but this is really a ruse for her to grab his gun. Her plan works, she takes the gun, and as Phillip tries to take it back, Vera shoots him dead.

The final line of the poem is: "One little Soldier Boy left all alone; He went out and hanged himself and then there were none." After killing Phillip, Vera is traumatized and overcome by guilt. She hangs herself according to the last line of the children's rhyme.

The police arrive on the island and attempt to make sense of the many mysterious deaths. They find a confession written by Justice Wargrave. In his confession, he says that he has always experienced, in equal measure, a desire for justice and bloodlust. He was able to satisfy both urges in his profession as a judge, but after being diagnosed with a terminal illness, he decided to punish several people who he felt escaped justice. He is responsible for the other characters' deaths and tricked Dr. Armstrong into helping him fake his death. After his plan is complete, he shoots himself, this time dying for real.

Anthony Marston is the first of the guests on the island to die. He dies of poisoning after someone spikes his drink with cyanide. Then it's the turn of Mrs. Rogers, who kicks the bucket in the middle of the night. Dr. Armstrong suspects that Mrs. Rogers' death is attributable to an overdose of sleeping tablets.

Under the circumstances, it's not surprising that a certain amount of fear descends upon the surviving guests. Indeed, General MacArthur, for one, is not just scared but positively fatalistic. He expresses the belief that no one will leave the island alive.

Well he certainly doesn't. At lunch one day, he's found dead with a heavy blow to the back of his skull. As with all the murders in the story, his takes place in accordance with the words of the song about the Ten Little Indian Boys. In the case of General MacArthur, he's the one who “said he'd stay there”.

The next to book a one-way ticket to eternity is Mr. Rogers, who's found dead the next day in the woodshed having been struck with an axe. Later on in the day, Emily Brent ends up dead in the kitchen from an injection of potassium cyanide.

Justice Wargrave is killed by a gunshot wound to the head. (Or so it appears, at any rate.) Blore has his skull smashed in by a clock; Armstrong is drowned; and Lombard is shot on the beach by Vera. Unable to live with herself, Vera commits suicide by hanging.

As stated very well above, the characters in Agatha Christie's book die according to the way the characters die in the song "Ten Little Indians." To provide a broader context, the characters in the mystery have been chosen and lured to an uninhabited island by "U.N. Owen" (his name means "unknown"). They cannot escape to shore because of bad weather and the distance from the mainland. Two of the victims are servants who have been recently hired. They have all been brought to the island because they committed crimes which, whether legal or not, were immoral and led to another person's wrongful death.

For example, Emily Brent, a religious hypocrite, fired a pregnant servant, Beatrice Taylor, for immorality. As a result, Beatrice drowned herself. What Brent did was not technically illegal, but it was cruel and hard-hearted. In another example, the butler and housekeeper, Mr. and Mrs. Roger, withheld medicine from an elderly woman once in their care, so that she would die. They then inherited money she left them in her will. Dr. Armstrong operated on a woman while he was drunk, killing her through his irresponsible behavior.

In other words, these are all people who, in the eyes of the murderer dispensing justice, deserve to die. 

They die in accordance with the song. The play was originally called "Ten Little Niggers", but, for obvious reasons, was then renamed "Ten Little Indians". When that too became politically unacceptable the play was renamed "And Then There Were None".

Here's the first bit of the song (full link below):

Ten little Indian boys went out to dine
One choked his little self and then there were nine
Nine little Indian boys sat up very late
One overslept himself and then there were eight
Eight little Indian boys traveling in Devon
One said he'd stay there and then there were seven...

Here's the deaths:

Anthony Marston is poisoned with cyanide in his drink ("one choked his little self")
Mrs. Rogers never wakes up rom her sleep ("one overselpt himself")
General MacArthur dies from a blow to the back of the head, after he said he'd never leave ("one said he'd stay there)
Thomas Rogers is struck with an axe in the woodshed ("one chopped himself in halves")
Emily Brent dies from a lethal injection ("a bumblebee stung one")
Judge Wargrave dies from a shot in the head ("one got into Chancery" - and then finally kills himself)
Blore dies by being crushed by a clock ("a big bear hugged one")
Armstrong is drowned in the sea ("a red herring stung one")
Lombard is shot with his own revolver by Vera ("out in the sun" or "one shot the other")
Vera then hangs herself ("one hanged himself").

Hope it helps!