Describe the Lukan account of the annunciation of the birth of Jesus; and how this account is different from Mathew's

According to St. Luke’s account, Angel Gabriel visited the
virgin, Mary, who was betrothed to Joseph in Nazareth. The angel hailed
(greeted) Mary as the favoured one and told her that the Lord was with her.
Mary was troubled in mind because she did not quite understand that manner of
greeting. The angel asked her not to be afraid, but told her that she would
conceive in her womb and bear a son to be called Jesus -The son of the Most
High. The Lord God would give him the throne of David and he would reign
forever.

Mary asked the angel how it would happen for she had no husband.
The angel told her that the Holy Spirit would come upon her and the power of
the most High would overshadow her, hence the child would be called Holy, the
son of God.

He confirmed to Mary that her cousin, Elizabeth had also conceived
a son in her old age. She was then six months’ pregnant. Though she had been
barren before then, with God, nothing was impossible. Mary said that she was
God’s handmaid and wished the angel’s words be fulfilled as prophesied. 

The
Luke account differs from that of Matthew in some ways: 

• In Matthew, an
  angel did not visit Mary, it was Joseph who saw an angel in a dream. The angel
  is not named, 
• Matthew alone mentions Joseph’s earlier intent to
  divorce Mary when she was found to be pregnant. There was no such information
  in Luke’s account, 
• Angel Gabriel spoke to Mary in St. Luke and she
  obeyed the word of God with faith, 
• In Matthew, there was no
  reference to Elizabeth as in Luke.