What does Luke 1:44-49 mean?

Thank you for your question about these inspiring verses in Luke, especially timely in this season of celebrating the birth of Jesus.  These events take place after the angel told Mary she would conceive a child through the Holy Spirit and that her cousin Elizabeth, advanced in years, had conceived a child.  Mary left Nazareth for a visit to Elizabeth and Elizabeth tells Mary that her baby leaped with joy when it heard Mary's voice – and, guided by the Holy Spirit, Elizabeth adds that Mary is surely blessed because she believed the angel's words to her.  

We can imagine the rejoicing these two women would have shared – these two holy women whom God chose to carry out His plan for mankind's salvation.  All in Israel were expecting the birth of the Messiah, the Savior, and it was Mary who was chosen to bear God's son.  What joy and awe must have been within Mary as she carried the child.  When the angel told her, she humbly answered, "Behold the handmaid of the Lord; be it unto me according to thy word."  Luke 1:38.  Mary's humility was a virtue that God was looking for in the one who would bear His son.  Mary's joy is expressed in the rest of the verses from your question.  
Luke 1:46  And Mary said, My soul does magnify the Lord,  (47)And my spirit has rejoiced in God my Savior.
Mary praises God, putting her complete trust in Him with all that she has (her body) and with all that she is (her being).   God is first, and she does not fear what others may say.  What a praise to God this is – to be completely peaceful in carrying out His will for her.  Her spirit is lifted heavenward toward God.  She loves God and rejoices in Him.
Luke 1:48  For he has regarded the low estate of his handmaiden:  Despite her humble station in life, God has chosen her to give birth to the Messiah, the King of Kings who will rule over earth forever in His holy kingdom.  For, behold, from henceforth all generations shall call me blessed. All generations, or ages throughout time, will know this, and she will be called blessed – all will say she is fortunate to have had this blessed privilege of giving birth to the Messiah.  
Luke 1:49  For he that is mighty has done to me great things; God had done a magnificent  favor to Mary for all to see and know.  Mary realized what a great privilege it was for the Lord to choose her as the mother of Jesus, the Messiah, and holy is his name. God's name is to be held sacred in His purity, majesty, and glory.  Mary praises God by submitting to Him, and His name is holy, sacred, and should be revered.

In these verses, Mary shines forth as a beautiful example of trust in God and humility.   Each of her words – magnify, rejoiced, blessed, holy is his name – help us see into her character of joy, love, submission, gentleness.  Today we look to Mary as a faithful follower of our Lord Jesus and as an example of one who completely did the Heavenly Father's will.  What true joy and peace this brings. 

Was James, who was the real brother of Jesus, the author of the book of James?

There are three men of importance named James in the New Testament.   First, there was the Apostle James, who was one of the “sons of Zebedee” (Matthew 4:21).  This James was beheaded by Herod (Acts 12:1-2). Second, James the son of Alphaeus (Mark 3:18), also was an Apostle, but he was not prominent in the New Testament books.  The third James (Matthew 13:55) would have been the natural child of Mary and Joseph and thus the “Lord’s brother” since they shared a common mother.  Initially, the Lord’s family did not believe in Jesus as the Messiah (John 7:5). However, they were eventually converted to Christianity as recorded in Acts 1:14.  This James became an important figure in the early church by both accepting Paul as an apostle and then by opposing Paul respecting Gentile conversions (Galatians 2:9-12). Paul also wrote, “Other of the apostles saw I none, save James the Lord’s brother.” [Galatians 1:19] The Greek for “save” means, “unless you want to consider” this James as one of the twelve Apostles, which he was not.

The history of this James, who wrote the epistle, is related by Eusebius in his Ecclesiastical History, Book 2, Chapter 1, written after 325 A.D.  He quotes the writings of the early Christian martyr Clement: “Then James, whom the ancients surnamed the Just on account of the excellence of his virtue, is recorded to have been the first to be made bishop [overseer] of the church of Jerusalem. This James was called ‘the brother of the Lord’ because he was known as a son of Joseph, and Joseph was supposed to be the father of Christ, because the Virgin, being betrothed to him, ‘was found with child by the Holy Ghost before they came together,’ as the account of the holy Gospels shows.  But Clement in the sixth book of his Hypotyposes writes thus: ‘For they say that Peter and James and John after the ascension of our Saviour, as if also preferred by our Lord, strove not after honor, but chose James the Just bishop [overseer] of Jerusalem.’  But the same writer, in the seventh book of the same work, relates also the following things concerning him: ‘The Lord after his resurrection imparted knowledge to James the Just and to John and Peter, and they imparted it to the rest of the apostles, and the rest of the apostles to the seventy, of whom Barnabas was one. But there were two Jameses: one called “the Just,” who was thrown from the pinnacle of the temple and was beaten to death with a club by a fuller, and another who was beheaded.’ ” 

Thus, by tradition and by the scriptures, it seems likely that James, the Lord’s natural brother, was the author of the book of James.