Sources

The pre-eminent source is the Gospel of St Luke, Ch. 1 v 26 - 38:

And in the sixth month the angel Gabriel was sent from God unto a city of Galilee, named Nazareth, to a virgin espoused to a man whose name was Joseph, of the house of David; and the virgin's name was Mary. And the angel came in unto her, and said, Hail, thou that art highly favoured, the Lord is with thee: blessed art thou among women. And when she saw him, she was troubled at his saying, and cast in her mind what manner of salutation this should be.
    And the angel said unto her, Fear not, Mary, for thou hast found favour with God. And, behold, thou shalt conceive in thy womb, and bring forth a son; and shalt call his name Jesus.
     He shall be great, and shall be called the Son of the Highest; and the Lord God shall give unto him the throne of his father David; and he shall reign over the house of Jacob for ever, and of his kingdom there shall be no end.
   Then said Mary unto the angel: How shall this be, seeing I know not a man?
   And the angel answered and, said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that Holy thing which shall be born of thee shall be called the Son of God.
    And behold, thy cousin Elisabeth, she hath also conceived a son in her old age: and this is the sixth month with her, who was called barren.
     For with God nothing shall be impossible.
    And Mary said, Behold the handmaid of the Lord; be it unto me according to thy word. And the angel departed from her.
  (Authorised Version)

Matthew Ch 1 v18 - 21 has a shorter version.

The medieval reader wanted more than this: a little more was supplied in these two chapters from the apocryphal text known as the Protoevangelium of James.

   10. And there was a council of the priests, saying: Let us make a veil for the temple of the Lord. And the priest said: Call to me the undefiled virgins of the family of David. And the officers went away, and sought, and found seven virgins. And the priest remembered the child Mary, that she was of the family of David, and undefiled before God. And the officers went away and brought her. And they brought them into the temple of the Lord. And the priest said: Choose for me by lot who shall spin the gold, and the white, and the fine linen, and the silk, and the blue, and the scarlet, and the true purple. And the true purple and the scarlet fell to the lot of Mary, and she took them, and went away to her house. And at that time Zacharias was dumb, and Samuel was in his place until the time that Zacharias spake. And Mary took the scarlet, and span it.

   11. And she took the pitcher, and went out to fill it with water. And, behold, a voice saying: Hail, thou who hast received grace; the Lord is with thee; blessed art thou among women! And she looked round, on the right hand and on the left, to see whence this voice came. And she went away, trembling, to her house, and put down the pitcher; and taking the purple, she sat down on her seat, and drew it out. And, behold, an angel of the Lord stood before her, saying: Fear not, Mary; for thou hast found grace before the Lord of all, and thou shalt conceive, according to His word. And she hearing, reasoned with herself, saying: Shall I conceive by the Lord, the living God? and shall I bring forth as every woman brings forth? And the angel of the Lord said: Not so, Mary; for the power of the Lord shall overshadow thee: wherefore also that holy thing which shall be born of thee shall be called the Son of the Most High. And thou shalt call His name Jesus, for He shall save His people from their sins. And Mary said: Behold, the servant of the Lord before His face: let it be unto me according to thy word.

  Another apocryphal text, The Gospel of Pseudo-Matthew, adds a nice story about the spitefulness of the other virgins, and an earlier visit of the angel:

   Then Joseph received Mary, with the other five virgins who were to be with her in Joseph's house. These virgins were Rebecca, Sephora, Susanna, Abigea, and Cael; to whom the high priest gave the silk, and the blue,  and the fine linen, and the scarlet, and the purple, and the fine flax. For they cast lots among themselves what each virgin should do, and the purple for the veil of the temple of the Lord fell to the lot of Mary. And when she had got it, those virgins said to her: Since thou art the last, and humble, and younger than all, thou hast deserved to receive and obtain the purple. And thus saying, as it were in words of annoyance, they began to call her queen of virgins. While, however, they were so doing, the angel of the Lord appeared in the midst of them, saying: These words shall not have been uttered by way of annoyance, but prophesied as a prophecy most true. They trembled, therefore, at the sight of the angel, and at his words, and asked her to pardon them, and pray for them.

   And on the second day, while Mary was at the fountain to fill her pitcher, the angel of the Lord appeared to her, saying: Blessed art thou, Mary; for in thy womb thou hast prepared an habitation for the Lord. For, lo, the light from heaven shall come and dwell in thee, and by means of thee will shine over the whole world.

   Again, on the third day, while she was working at the purple with her fingers, there entered a young man of ineffable beauty. And when Mary saw him, she exceedingly feared and trembled. And he said to her: Hail, Mary, full of grace; the Lord is with thee: blessed art thou among women, and blessed is the fruit of thy womb. And when she heard these words, she trembled, and was exceedingly afraid. Then the angel of the Lord added: Fear not, Mary; for thou hast found favour with God: Behold, thou shalt conceive in thy womb, and shalt bring forth a King, who fills not only the earth, but the heaven, and who reigns from generation to generation.

These references from the Old Testament are claimed to refer to the Annunciation:

Isaiah, Ch 7.  
Here ye now, O house of David; Is it a small thing for you to weary men, but will ye weary my God also?
Therefore the Lord himself shall give you a sign; behold, a virgin shall conceive, and bear a son, and shall call his name Immanuel.

Song of Solomon Ch 4
12 A garden inclosed is my sister, my spouse; a spring shut up, a fountain sealed.
13 Thy plants are an orchard of pomegranates, with pleasant fruits; camphire, with spikenard.
14 Spikenard and saffron; calamus and cinnamon, with all trees of frankincense; myrrh and aloes, with all the chief of spices:
15 A fountain of gardens, a well of living waters, and streams from Lebanon.
16 Awake, O north wind; and come, thou south; blow upon my garden, that the spices thereof may flow out. Let my beloved come into his garden, and eat his pleasant fruits.

The Golden Legend provides a detailed interpretation of the events. The recent translation by  William Granger Ryan (Princeton University Press) is recommended, though earlier versions are available on the internet.
Another version of the story is found in the Koran, though, for fairly obvious reasons, this version would not have influenced artists.