The following is an annotated version of the fairy tale. I recommend reading the entire story before exploring the annotations, especially if you have not read the tale recently.

MOST terribly cold it was; it snowed, and was nearly quite dark, and evening-- the last evening of the year.1 In this cold and darkness there went along the street a poor little girl, bareheaded, and with naked feet. When she left home she had slippers2 on, it is true; but what was the good of that? They were very large slippers, which her mother had hitherto worn; so large were they; and the poor little thing lost them as she scuffled away across the street, because of two carriages that rolled by dreadfully fast.

One slipper was nowhere to be found; the other had been laid hold of by an urchin, and off he ran with it; he thought it would do capitally for a cradle when he some day or other should have children himself. So the little maiden walked on with her tiny naked feet, that were quite red and blue from cold. She carried a quantity of matches in an old apron, and she held a bundle of them in her hand. Nobody had bought anything of her the whole livelong day; no one had given her a single farthing.3

She crept along trembling with cold and hunger4--a very picture of sorrow, the poor little thing!

The flakes of snow covered her long fair hair, which fell in beautiful curls around her neck; but of that, of course, she never once now thought. From all the windows the candles were gleaming, and it smelt so deliciously of roast goose, for you know it was New Year's Eve; yes, of that she thought.

In a corner formed by two houses, of which one advanced more than the other, she seated herself down and cowered together.5 Her little feet she had drawn close up to her, but she grew colder and colder, and to go home she did not venture, for she had not sold any matches and could not bring a farthing of money: from her father she would certainly get blows,6 and at home it was cold too, for above her she had only the roof, through which the wind whistled, even though the largest cracks were stopped up with straw and rags.7

Her little hands were almost numbed with cold. Oh! a match might afford her a world of comfort, if she only dared take a single one out of the bundle, draw it against the wall, and warm her fingers by it.8 She drew one out. "Rischt!" how it blazed, how it burnt! It was a warm, bright flame, like a candle, as she held her hands over it: it was a wonderful light. It seemed really to the little maiden as though she were sitting before a large iron stove, with burnished brass feet and a brass ornament at top. The fire burned with such blessed influence; it warmed so delightfully. The little girl had already stretched out her feet to warm them too; but--the small flame went out, the stove vanished: she had only the remains of the burnt-out match in her hand.9

She rubbed another against the wall: it burned brightly, and where the light fell on the wall, there the wall became transparent like a veil, so that she could see into the room. On the table was spread a snow-white tablecloth; upon it was a splendid porcelain service, and the roast goose was steaming famously with its stuffing of apple and dried plums. And what was still more capital to behold was, the goose hopped down from the dish, reeled about on the floor with knife and fork in its breast,10 till it came up to the poor little girl; when--the match went out and nothing but the thick, cold, damp wall was left behind. She lighted another match. Now there she was sitting under the most magnificent Christmas tree:11 it was still larger, and more decorated than the one which she had seen through the glass door in the rich merchant's house.

Thousands of lights were burning on the green branches, and gaily-colored pictures, such as she had seen in the shop-windows, looked down upon her. The little maiden stretched out her hands towards them when--the match went out. The lights of the Christmas tree rose higher and higher, she saw them now
as stars in heaven; one fell down and formed a long trail of fire.

"Someone is just dead!"12 said the little girl; for her old grandmother, the only person who had loved her,13 and who was now no more, had told her, that when a star falls, a soul ascends to God.14

She drew another match against the wall: it was again light, and in the lustre there stood the old grandmother, so bright and radiant, so mild, and with such an expression of love.15

"Grandmother!" cried the little one. "Oh, take me with you! You go away when the match burns out; you vanish like the warm stove, like the delicious roast goose, and like the magnificent Christmas tree!" And she rubbed the whole bundle of matches quickly against the wall, for she wanted to be quite sure of keeping her grandmother near her. And the matches gave such a brilliant light that it was brighter than at noon-day: never formerly had the grandmother been so beautiful and so tall. She took the little maiden, on her arm, and both flew in brightness and in joy so high, so very high, and then above was neither cold, nor hunger, nor anxiety--they were with God.16

But in the corner, at the cold hour of dawn, sat the poor girl, with rosy cheeks and with a smiling mouth, leaning against the wall--frozen to death on the last evening of the old year. Stiff and stark sat the child there with her matches, of which one bundle had been burnt. "She wanted to warm herself," people said. No one had the slightest suspicion of what beautiful things she had seen; no one even dreamed of the splendor in which, with her grandmother she had entered on the joys of a new year.17

1. Last evening of the year: New Year's Eve, as mentioned specifically later in the story. The Christmas and New Year holidays are known for great times of charitable giving. This story has become one of the most popular tales for inspiring charitable donations, especially during the Christmas holidays. It's also a reminder that this little girl, like so many others in poverty, needs help every day of the year, not just Christmas.
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2.  Slippers: The son of a cobbler, Andersen shows great interest in shoes and slippers in many of his tales, as well as feet such as in the Little Mermaid.
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3.  Nobody had bought anything of her the whole livelong day; no one had given her a single farthing: Begging was illegal during Andersen's time. The poor would make matches and sell them on the street as a front for their actual begging.
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4.  She crept along trembling with cold and hunger: Shivering is one of the early symptoms of hypothermia. It is muscle activity generated by the body to produce heat. As the body cools down further, shivering will stop.

You can read more about hypothermia, its symptoms and treatment at WebMD.com and NatureSkills.com.

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5.  In a corner formed by two houses, of which one advanced more than the other, she seated herself down and cowered together: Sitting down to rest in the cold is the worst decision to make when threatened with freezing to death, medically known as hypothermia. Movement and exercise helps the body retain heat. Sitting quickens the onset of hypothermia and consequent death.
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6.  From her father she would certainly get blows: Not only is the child cold and hungry, she is also abused at home, increasing the pathos and stark reality of the story.
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7.  Largest cracks were stopped up with straw and rags: Straw and rags were often used for insulation and building materials in the ill-made shelters of the poor.
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8. Oh! a match might afford her a world of comfort, if she only dared take a single one out of the bundle, draw it against the wall, and warm her fingers by it: The matches are her wares to be sold, not to be used by herself despite her misery. Surely using the matches without receiving money would cause another beating at home.
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9. The stove vanished: she had only the remains of the burnt-out match in her hand: Vision number one. This vision directly addresses the girl's coldness.

As the little girl begins to freeze, she experiences four visions. Some might consider all four to be hallucinations, since hallucinations are a common symptom of hypothermia. While Andesen probably considered the first three visions to be hallucinations, his religious beliefs insure his intention that the final vision of the grandmother be a real experience.
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10.  The goose hopped down from the dish, reeled about on the floor with knife and fork in its breast: Vision number two. This vision directly addresses the girl's hunger. Goose is a traditional meal served during the Christmas and New Year holidays, but one the matchgirl's family could ill afford.

The image of the goose dancing with knife and fork in its breast may be horrifying to some sensibilities but cartoonish to others. Andersen's intent is unclear.
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11. The most magnificent Christmas tree: Vision number three. While this vision does not address any of the girl's physcial needs, it reinforces the tale's Christian themes and foreshadows the girl's vision of her grandmother in heaven that follows. It also provides the device for seeing the shooting star in the sky.

You can read more about the history of the Christmas Tree, which was in use during Andersen's lifetime, on the Christmas Tree Farm Network and The Christmas Archives.

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12.  "Someone is just dead!": While the shooting star foreshadows her own death, it is debatable that this particular star represents the girl due to its timing in the story.
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13.  Her old grandmother, the only person who had loved her: Andersen also had a grandmother who doted on him, whom he remembered fondly in his memoirs and honors in this tale.
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14.  When a star falls, a soul ascends to God: A folklore superstition. A corollary superstition states that a shooting star represents a soul escaping purgatory.

A similar Creole superstition states: "Shooting-stars are souls escaping from purgatory: if you can make a good wish three times before the star disappears, the wish will be granted." NEW ORLEANS SUPERSTITIONS by Lafcadio Hearn

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15.  There stood the old grandmother, so bright and radiant, so mild, and with such an expression of love: The fourth and final vision.

Many near-death experiences around the world, regardless of religious belief, involve the visitation of dead loved-ones, usually family members and close friends.
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16.  Both flew in brightness and in joy so high, so very high, and then above was neither cold, nor hunger, nor anxiety--they were with God: Andersen's religious beliefs included a firm belief that the innocent enter God's presence at death where there is no more suffering.
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17.  Frozen to death on the last evening of the old year . . . No one had the slightest suspicion of what beautiful things she had seen; no one even dreamed of the splendor in which, with her grandmother she had entered on the joys of a new year: This is Andersen's version of a happy ending. The little girl is with God and her grandmother, never to suffer more in her worldly existence. Even being rescued from her plight and escaping death would not be as wonderful to him as the ending he chose.

Despite Andersen's obvious intentions, many translators and publishers of the tale have balked against this ending, occasionally offering a "happier" ending with the young girl being rescued by a family, to be fed and warmed on New Year's Eve with the implication that all would be well with her in the coming year.

Either ending serves another great purpose: reminding people to be charitable and help the poor during the holidays, and hopefully year round, to keep young children from suffering with poverty and death.
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