Who, exactly, are James son of Alphaeus and Judas the Son [Brother] of James?

Well they were all brothers of Christ even Peter, but Bartholomew is known to have instantly have faith in him after his initial prejudice. Meaning he thought Christ would have been a different race. As for Judas being James blood brother this is also true, but Judas carried the curse and James got the blessing. Similar to Cain and Able. Obviously they both got the curse of death but Judas got the worst of it by suicide- James fell like Jesus to the hands of capital punishment.

The book of James and Jude are the siblings of Christ. James son of Alpheus is the father of a Judah aka Thaddaeus Lebbaeus who followed along with his father. Whether Matthew and this James are brothers is up for debate because they are never mentioned as brethren so it is possible there are two Alpheus. The names were so popular and common at that time if you were not physically there it is easy to confuse one for another on paper. As far as the men being teenagers this is also for debate because of a scene where a synangogue tax collecter talked to Peter about Jesus not paying a specific tax. Does that mean the rest were under 20 or does it mean most of them paid and this scene was specifically towards Peter concerning his Rabbi? When Yehoshua told John to take care of his mother that implies that she had no other sons because in Hebrew culture it would be the duty of the next son after the eldest had deceased to take care of his parents. The mention of his siblings are after the mention of Joseph, Joseph's trade and Joseph's children who of course would be called his brothers because their father and his mother were married.Joseph was not a young boy when Mariam and himself were espoused he was already up in age infact he was ten years older than King Herod so by the time Yehoshua was born the man Joseph was in his late 50s with grown children. The fact that his brothers did not believe in him and also jeered at him to go up to Jerusalem shows you they were older and teasing their younger sibling who they felt was besides himself. He was Mary's only son out her womb but she had other legal sons and they were older than him.

According to the Bible, there was a tax collector named Matthew (also called Levi) son of Alphaeus. James son of Alphaeus is most likely Matthew's brother.

As said in the comments, there were many people by the names of Jude and of James. Jude could have been anybody; and if Jude is the son of James, his dad was not a disciple nor a brother of Jesus. Most of the Disciples were as old as teenagers, with the exception of Peter. Only Peter and Jesus had to pay taxes (Matthew 17:24-27). Jude couldn't have been a brother of Jesus because none of them believed He was the Messiah until after the resurrection. Greek translations show that Jude was called Jude James. In the Israelite culture, sons would inherit their father's name. So James was probably Jude's dad.

Modern Protestants generally believe that there are five individuals named James in the New Testament.

1. James the brother of Jesus, eventual leader in Church of Jerusalem (Mark 6:3, Matthew 13:55, Acts 12:17, Acts 15:12-23, Acts 21:17-24, Galatians 2:7-9, 1 Corinthians 15:3-8, Galatians 1:19, Jude 1).
2. James the son of Zebedee, an Apostle (Luke 6:13-16, Mark 3:16-19, Matthew 17:1-2, Matthew 26:36-38, died in Acts 12:1-2)
3. James the son of Alphaeus, an Apostle (Matthew 10:1-4, Mark 3:16-19, Luke 6:13-16, Acts 1:13)
4. James the father of the Apostle Judas (not Iscariot) (Luke 6:13-16, Acts 1:13)
5. James the Less, son of the "other Mary" (Mark 15:40, Matthew 27:56)

Note that there is some variance - for example, John MacArthur (evangelical pastor, author of The MacArthur Bible Commentary) says in his commentary on Matthew 10:3 that James the Less and James the son of Alphaeus are the same James.

The early Church Fathers, however, generally believed there were only two individuals named James in the New Testament - James the Greater and James the Less.

James the Greater

• Apostle, son of Zebedee (Luke 6:13-16, Mark 3:16-19)
• He and his brother John (also an Apostle) were given the nicknames Sons of Thunder by Christ (Mark 3:16-19)
• His mother Salome was one of the women who followed Jesus, ministering to him, and was there when Jesus died (Matthew 27:55-56, Mark 15:40, Mark 16:1).
• James, his brother John, and Peter were part of the inner circle of Christ - the group of three Apostles with whom Jesus was especially close, that saw him transfigured on the mountain, and with whom Jesus shared his sorrow in the garden of Gethsemane (Matthew 17:1-2, Matthew 26:36-38)
• It is because of this closeness with Christ that we call him "James the Greater", to distinguish him from the other James, who was called in Scripture "James the Less" (implying a comparison between two people named James).
• He died by Herod's hand in the book of Acts (Acts 12:1-2) around 44 AD.

James the Less

• Son of the "other Mary", and brother of Joseph (Mark 15:40, Matthew 27:56)
• This "other Mary" was one of the women who followed and ministered to Jesus, and was looking on the cross from a distance as Jesus died.
• In Matthew, Mark, and Luke she is identified as "Mary the mother of James the less and of Joses" (Mark 15:40), "Mary the mother of Joses" (Mark 15:47), "Mary the mother of James" (Mark 16:1, Luke 24:10), "Mary the mother of James and Joseph" (Matthew 27:56), "Mary Magdalene and the other Mary" (Matthew 27:61, 28:1).
• In the Gospel of John she is identified as "Mary of Clopas", sister (adelphe) of Mary the mother of Jesus (John 19:25).
• Eusebius (a church historian around 323 AD), quotes an older Chronicler named Hegesippus who died in 180 AD in his Church History Book III, Chapter 11, saying "For Hegesippus records that Clopas was a brother of Joseph."
• So this "other Mary" is married to Clopas, brother of Joseph - making her sister-in-law to Mary the mother of Jesus. She is Jesus' aunt, and her children are Jesus' cousins.

Who are her children?

Matthew, Mark, and Luke mention two when identifying who she was near the cross - James the less and Joseph. In addition to those two, Matthew and Mark tell us of Judas and Simon as well (Matthew 13:55, Mark 6:3). So Jesus had four named cousins ('adelphos') in Scripture: James the less, Joseph, Judas, and Simon.

This means James the Less is the 'adelphos' (cousin) of Christ!

James the 'adelphos' of Christ wrote the New Testament Epistle of James, and became a leading figure in the church at Jerusalem - when the church met in a council to determine rules for Gentile Christians, James played a prominent role in the decision and laid down his judgment (Acts 15:12-13, 19-23).

Paul identified this James as an Apostle (Galatians 1:19), and called him (along with Cephas and John) the pillars of the church (Galatians 2:7-9). Paul regularly went out of his way to show deference to this James and sought his counsel (Acts 12:17, Acts 21:17-24, 1 Corinthians 15:3-8, Galatians 1:19).

Jerome (~393 AD), quoting Hegesippus (~180 AD), identifies James the 'adelphos' of the Lord as James the Just, head of the Church at Jerusalem (On Illustrious Men, Section 2).

Backing up a bit, note how Paul identified this James as an Apostle in Galatians 1:19. Which Apostle was he?

Looking at church history, we see that James the 'adelphos' of Christ and the Apostle James the son of Alphaeus both died the same way - via stoning by the Jews in Jerusalem (Pseudo-Hippolytus in On the Twelve Apostles, and Josephus in Antiquities of the Jews, Book XX, Chapter 9, Section 1).

If this James is called James the Less, it stands to reason he is James the son of Alphaeus in the lists of Apostles - as James the son of Zebedee, being in the inner circle of Christ, would be James the Greater.

The issue arises, how can he be the son of Alphaeus, if we have already identified him as the son of Mary, wife of Clopas?

There are several theories that plausibly explain this.

1) Mary could be the wife of Alphaeus, not Clopas. The Greek text for her name simply says "Mary of Clopas", not "Mary the wife of Clopas". Most modern translations assume the relationship of wife. The title "of Clopas" could have been given to her after her father, or kindred, or for some other reason. This theory is articulated in the Fragment of Papias X from ~100 AD, and in Jerome's work Against Helvidius from around 383 AD.

2) The man could have had two names, Alphaeus and Clopas. As Jerome notes around 383 AD, this is not an uncommon occurrence in Scripture. Jerome gives numerous examples in his text Against Helvidius: "Raguel, Moses' father-in-law, is also called Jethro. Gedeon, without any apparent reason for the change, all at once becomes Jerubbaal. Ozias, king of Judah, has an alternative, Azarias. Mount Tabor is called Itabyrium. Again Hermon is called by the Phenicians Sanior, and by the Amorites Sanir. The same tract of country is known by three names, Negebh, Teman, and Darom in Ezekiel. Peter is also called Simon and Cephas. Judas the zealot in another Gospel is called Thaddaeus. And there are numerous other examples which the reader will be able to collect for himself from every part of Scripture."

3) The name Clopas may be a Greek transliteration of an Aramaic Alphaeus.

At this point, we have identified James the 'adelphos' of Christ who became leader of the Church of Jerusalem (with the title of James the Just), James the Less the cousin of Christ, and the Apostle James of Alphaeus as all the same person. This James is the author of the Epistle of James in the New Testament.

Jude

Let us now concern ourselves with the brothers of this James - Joseph, Judas and Simon (Matthew 13:55, Mark 6:3).

This brother named Judas wrote the book of Jude, where he identifies himself in relationship to his brother James (Jude 1).

In the list of Apostles in Luke 6:13-16, we see a "Judas of James" listed. The modern ESV translation says "Judas the son of James", the 1611 King James Version says "Judas the brother of James", but the original Greek simply says "Judas of James" with the exact relationship unspecified. Only in one other spot does a Jude identify himself in relationship to his brother James, so we hold this relationship to be that of brother, and thus this Apostle Judas to be the cousin of Christ, brother of James the Less.

In the list of Apostles in Mark 3:16-19, we see that this Jude/Judas was also known by the name Thaddeus (likely to distinguish him from the traitor, Judas Iscariot).

So Jesus' cousin Judas also authored a book of the New Testament, the Epistle of Jude - and he was an Apostle known as Judas of James or Thaddeus.

The Fragments of Papias from around 100 AD supports the above theory:

Mary the wife of Cleophas or Alphaeus, who was the mother of James the bishop and apostle, and of Simon and Thaddeus, and of one Joseph... James and Judas and Joseph were sons of an aunt of the Lord's... Mary, mother of James the Less and Joseph, wife of Alphaeus was the sister of Mary the mother of the Lord, whom John names of Cleophas, either from her father or from the family of the clan, or for some other reason.

The Council of Rome (~382 AD) also identifies the Apostle Judas the Zealot as the author of the book of Jude, and thus the brother of James:

Now indeed the issue of the divine scriptures must be discussed, which the universal Catholic church receives or which it is required to avoid...

Four books of the Gospels: according to Mathew, according to Mark, according to Luke, according to John. Likewise the acts of the apostles. The letters of the apostle Paul in number fourteen: to the Romans, to the Corinthians two letters, to the Ephesians, to the Thesalonians two letters, to the Galatians, to the Philippians, to the Colossians, to Timothy two letters, to Titus, to the Philemon, to the Hebrews. Likewise the apocalypse of John. Likewise the canonical [catholic] letters in number seven: of the apostle Peter two letters, of the apostle James one letter, of the apostle John one letter, of the other John the elder two letters, of the apostle Judas the Zealot one letter.

HERE ENDS THE CANON OF THE NEW TESTAMENT.

So in summary, James the son of Alphaeus and Judas the brother of James were both Jesus' cousins, born of Mary of Clopas, who was sister (in-law) to Mary the mother of Jesus. Both were apostles, and James was a leader of the church of Jerusalem.

Additional information and references on this are available here.