Who, exactly, are James son of Alphaeus and Judas the Son [Brother] of James? One note specifically is that some translations read that Judas (not Iscariot) was known as Judas brother of James, but most say Judas Son of James. 

Luke 6:16 NIV


  Judas son of James, and Judas Iscariot, who became a traitor.


Compare that to:

Luke 6:16 KJV


  And Judas the brother of James, and Judas Iscariot, which also was the traitor.


So, who were the apostles named Judas and James son of Alphaeus, exactly? If these apostles were brothers, wouldn't they share the same father? James is known as James son of Alphaeus...

Luke 6:15 NIV


  Matthew, Thomas, James son of Alphaeus, Simon who was called the
  Zealot,


In Jude, the author also introduces himself as the brother of James 
Jude 1:1 NIV


  Jude, a servant of Jesus Christ and a brother of James,


However, this can also be Judas the brother of Christ and James, also the brother of Christ, but not necessarily the original apostles named Judas and James.

I ask because today at study, the individual giving the study was positing that Judas brother/son of James and James were both brothers of Christ, whereas I have it understood that none of Christ's brothers came to faith in him until the post-resurrection appearances.

Any help is appreciated!
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.