The word “disciple” refers to a learner or follower. The word “apostle” means “one who is sent out.” While Jesus was on earth, his twelve followers were called disciples. After his resurrection and ascension, Jesus sent the disciples out to be His witnesses (Matthew 28:18-20; Acts 1:8). They were then referred to as the twelve apostles. However, even when Jesus was still on earth, the terms “disciples” and “apostles” were used somewhat interchangeably.
The original twelve disciples/apostles are listed in Matthew 10:2–4: “These are the names of the twelve apostles: first, Simon (who is called Peter) and his brother Andrew; James son of Zebedee, and his brother John; Philip and Bartholomew; Thomas and Matthew the tax collector; James son of Alphaeus, and Thaddaeus; Simon the Zealot and Judas Iscariot, who betrayed Him.”
The Bible also lists the twelve disciples/apostles in Mark 3:16–19: “He appointed the twelve: Simon (to whom he gave the name Peter); James the son of Zebedee and John the brother of James (to whom he gave the name Boanerges, that is, Sons of Thunder); Andrew, and Philip, and Bartholomew, and Matthew, and Thomas, and James the son of Alphaeus, and Thaddaeus, and Simon the Zealot, and Judas Iscariot, who betrayed him.”
Here is the list of the original twelve in Luke 6:13–16: “And when day came, he called his disciples and chose from them twelve, whom he named apostles: Simon, whom he named Peter, and Andrew his brother, and James and John, and Philip, and Bartholomew, and Matthew, and Thomas, and James the son of Alphaeus, and Simon who was called the Zealot, and Judas the son of James, and Judas Iscariot, who became a traitor.”
A comparison of the three passages shows a couple of minor differences in the names. It seems that Thaddaeus was also known as “Judas, son of James” (Luke 6:16). Simon the Zealot was also known as Simon the Canaanite (Mark 3:18). The Gospel of John uses the name “Nathanael” instead of “Bartholomew,” but Nathanael and Bartholomew were undoubtedly the same person. Judas Iscariot, who betrayed Jesus, was replaced in the twelve apostles by Matthias (see Acts 1:20–26).
Luke 6:12-13 shows the 12 were all called by Jesus on the same day. Jesus spent the night in prayer prior to selecting his 12 apostles from his disciples.
Edit: John 1:35-50 sheds some light on the order in which Jesus first met his disciples. Andrew (son of John) first met Jesus then called his brother Simon (Peter) and said "we have found the Messiah." Then Philip (also a fisherman from Bethsaida) meets Jesus and tells Nathaniel (or Bartholomew).
Matthew, Mark and Luke mention James and John (sons of Zebedee) and Mark and Luke mention Levi, son of Alphaeus, the tax collector (Matthew)
That leaves Thomas, James (son of Alphaeus), Thaddaeus (or Judas the son of James) and Simon the Zealot (Simon the Canaanite) unaccounted for. Judas, son of Simon Iscariot, is mentioned in John 6:71.