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.
A POOR man had twelve children and was forced to work night and day to give             them even bread. When therefore             the thirteenth2 came into the world, he knew not what             to do in his trouble, but ran out into the great highway, and resolved             to ask the first person whom he met to             be godfather.3 The first to meet him was the good God             who already knew what filled his heart, and said to him, "Poor             man, I pity thee. I will hold thy child at its christening, and will             take charge of it and make             it happy on earth."4 The man said, "Who art             thou?" "I am God." "Then I do not desire to have             thee for a godfather," said the man; "thou givest to the rich,             and leavest the poor to hunger." Thus spoke the man, for he             did not know how wisely God apportions riches and poverty.5 He turned therefore away from the Lord, and went farther. Then the Devil             came to him and said, "What seekest thou? If thou wilt take me             as a godfather for thy child, I will give him gold             in plenty and all the joys of the world as well."6 The man asked, "Who art thou?" "I am the Devil."             "Then I do not desire to have thee for godfather," said the             man; "thou deceivest men and leadest them astray." He went             onwards, and then             came Death7 striding up to him with withered legs, and             said, "Take me as godfather." The man asked, "Who art             thou?" "I am Death, and I make all equal." Then said             the man, "Thou art the right one, thou             takest the rich as well as the poor, without distinction;8 thou shalt be godfather." Death answered, "I             will make thy child rich and famous,9 for he who has             me for a friend can lack nothing." The man said, "Next Sunday             is the christening; be there at the right time." Death appeared             as he had promised, and stood godfather quite in the usual way.
          
When the boy had grown up, his godfather one day appeared and bade him go with him. He led him forth into a forest, and showed him a herb which grew there,10 and said, "Now shalt thou receive thy godfather's present. I make thee a celebrated physician. When thou art called to a patient, I will always appear to thee. If I stand by the head of the sick man,11 thou mayst say with confidence that thou wilt make him well again, and if thou givest him of this herb he will recover; but if I stand by the patient's feet, he is mine, and thou must say that all remedies are in vain, and that no physician in the world could save him. But beware of using the herb against my will, or it might fare ill with thee."
It was not long before the youth was the most famous physician in the whole world. "He had only to look at the patient and he knew his condition at once, and if he would recover, or must needs die." So they said of him, and from far and wide people came to him, sent for him when they had any one ill, and gave him so much money that he soon became a rich man. Now it so befell that the King became ill, and the physician was summoned, and was to say if recovery were possible. But when he came to the bed, Death was standing by the feet of the sick man, and the herb did not grow which could save him. "If I could but cheat Death for once," thought the physician, "he is sure to take it ill if I do, but, as I am his godson, he will shut one eye; I will risk it." He therefore took up the sick man, and laid him the other way, so that now Death was standing by his head. Then he gave the King some of the herb,12 and he recovered and grew healthy again. But Death came to the physician, looking very black and angry, threatened him with his finger, and said, "Thou hast overreached me; this time I will pardon it, as thou art my godson; but if thou venturest it again, it will cost thee thy neck, for I will take thee thyself away with me."
Soon afterwards the King's daughter fell into a severe illness. She was his only child, and he wept day and night, so that he began to lose the sight of his eyes, and he caused it to be made known that whosoever rescued her from death should be her husband and inherit the crown. When the physician came to the sick girl's bed, he saw Death by her feet. He ought to have remembered the warning given by his godfather, but he was so infatuated by the great beauty13 of the King's daughter, and the happiness of becoming her husband, that he flung all thought to the winds. He did not see that Death was casting angry glances on him, that he was raising his hand in the air, and threatening him with his withered fist. He raised up the sick girl, and placed her head where her feet had lain. Then he gave her some of the herb, and instantly her cheeks flushed red, and life stirred afresh in her.
When Death saw that for a second time he was defrauded             of his own property,14 he walked up to the physician             with long strides, and said, "All is over with thee, and now the             lot falls on thee," and seized him so firmly with his ice-cold             hand, that he could not resist, and led him into a cave             below the earth.15 There he saw how thousands and thousands             of candles were             burning16 in countless rows, some large, others half-sized,             others small. Every instant some were extinguished, and others again             burnt up, so that the flames seemed to leap hither and thither in perpetual             change. "See," said Death, "these are the lights of men's             lives. The large ones belong to children, the half-sized ones to married             people in their prime, the little ones belong to old people; but children             and young folks likewise have often only a tiny candle." "Show             me the light of my life," said the physician, and he thought that             it would be still very tall. Death pointed to a little end which was             just threatening to go out, and said, "Behold, it is there."             "Ah, dear godfather," said the horrified physician, "light             a new one for me, do it for love of me, that I may enjoy my life, be             King, and the husband of the King's beautiful daughter." "I             cannot," answered Death, "one must go out before a new one             is lighted." "Then place the old one on a new one, that will             go on burning at once when the old one has come to an end," pleaded             the physician. Death behaved as if he were going to fulfill his wish,             and took hold of a tall new candle; but as he desired to revenge himself,             he purposely made a mistake in fixing it, and the little piece fell             down and was extinguished. Immediately the physician fell on the ground,             and now he             himself was in the hands of Death.17
              
Grimm, Jacob and Wilhelm. Household Tales. Margaret Hunt, translator. London: George Bell, 1884, 1892. 2 volumes.