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!
Justice Wargrave dies by shooting himself in the forehead(in the end, the one where he was found shot in the head was a fake), Philip Lombard dies by being shot through the head by Vera Claythorne, Miss Brent and Anthony Marston died from cyanide poisoning, Mrs. Rogers died from an overdose of chloral, Mr. Rogers' head was split open by a butcher's knife, Blore's head was crushed in, and Armstrong died of drowning.
Then were all brought to the Island for the purpose of being killed. The killer knows all the past discressions and believes they deserve to die. To keep the killer from being caught, he plots his own death and makes appear as if the others were killing each other instead, but in the end, he says how he is tossing a bottle with a note in it into the lake.