Luke continues his accounts of the preaching/teaching and healing ministry of Jesus as He travels. Remember that the "shape" of Luke's Gospel account, from a geography perspective, is sort of like an hourglass. The life and ministry of Jesus starts out "wide", with ministry beginning in a wide area in northern Israel—we hear in Luke 7 about places like Capernaum and Nain, which are in the north—and then "narrows" to Jerusalem before widening out again as the Gospel goes out to all the nations. Here in Luke 7, the account is "wide" in more than a geographic sense, as we hear about ministry not only among the people of Israel, but among Gentiles as well. Luke is showing us that Jesus has truly come for all people, not just the nation of Israel.
Some notes on the text specifically:
7:1 When Luke records that people are "hearing" Jesus' teaching, it's more than just a simple "listening". Those who were "hearers of the word" were what we might call catechumens, or students. The early Church (in the first three centuries after Jesus) would designate people as catechumens who were going through a three-year period of learning before being baptized. (That word group we use—such as catechism, catechesis, catechumen—they call come from the Greek word katēcheō which means "to share a communication that someone has received; to teach/instruct". To use the Greek parts of the word literally it means to "down-sound", or to "down-ring-out", having the sense of oral instruction that is passed down.)
7:2 A Roman centurion was a soldier in charge of a hundred other soldiers.
7:3 It's most curious that a Roman would send Jewish elders as messengers to a Jewish Rabbi! Remember that many of the Jewish people hated the Roman government. Yet this centurion seems to be well respected by the Jews, and that seems to go both ways.
7:4-5 The elders say to Jesus that the Roman is "worthy". They are claiming to Jesus that this man deserves whatever help he is seeking because has has shown kindness, generosity, and favour to the nation of Israel, most especially by using his wealth to fund a synagogue in Capernaum.
7:6-8 The centurion doesn't share the same assessment as the elders of his worthiness! As a Gentile, if he brought a Jewish person into his house, that Jewish person would be ritually unclean. He respects that as well as trusting that Jesus can work in the same way that he himself does: by a word. The centurion's faith in Jesus is such that he think Jesus can heal his servant in the same way that he commands his soldiers and they obey!
7:9 Jesus marvels at this man's faith and commends him, even comparing his faith to Israel and remarking that the people of Israel don't exhibit the same faith. This undoubtedly created some tension among the crowds He was teaching, among those who were following Him.
7:11-17 Not only does Jesus heal people of sickness, He even is able to raise the dead! This is a whole other level in the people's eyes, and because of miracles like this, word about him spread quickly around the whole area.
7:16 Notice the connection between the physical world and theology here. Healing (and resurrection) was connected to prophecy; it had to be the word of God accomplishing these great things.
7:18 At this point, John is in prison for the word that he had been speaking against Herod (Luke 3:19-20). Just as reports went out about Jesus in general, John's own disciples/catechumens reported to him.
7:19 "The one who is to come" or more literally "the coming one", similar to "hearer of the word" was more of a technical title. "The Coming One" was the promised Messiah.
7:23 John's inquiry of Jesus (again through messengers, just like with the centurion) is answered by Jesus with a reference to the prophecies of Isaiah 35 and 61, and other places in the Old Testament. These are being fulfilled by Jesus, and so He's proving that He is indeed the Messiah.
7:30 The significance of this note may too easily escape us: those who rejected John's baptism were really rejecting God's purpose for them. They were rejecting what God had to offer them.
7:33-34 Here's a "theological no-win situation". John avoids things commonly seen as vices and is rejected. Jesus participates in feasts—He's not like John in that regard—and is rejected. Again here we also see a bit of a technical term; "tax collectors and sinners" were worse than regular "sinners" in the eyes of many. We might use the term "capital-S Sinner" when we think of how this group of people was thought of.
7:36 There's no small amount of irony that Jesus has just called out the Pharisees' hypocrisy over things like meals and then goes to a feast at one of their houses!
7:37 This woman is put into that category of capital-S Sinner. In 7:39, the Pharisee is thinking to himself of the ritual uncleanness that Jesus would experience by coming into physical contact with this woman. As with healings of things like leprosy, Jesus reverses things: He doesn't become unclean, but the person He has contact with becomes clean!
7:37-38 Because Jesus specifically was there, she was moved to do this anointing with oil. Anointing with oil was a common religious ritual, connected in the Old Testament especially with being chosen as a leader (e.g. 1 Samuel 10:1). It's also interesting that "Messiah" is not only "The Coming One", but also "The Anointed (that is, Chosen) One".
7:41-47 Jesus often used parables to teach. This one specifically he used to bring a word of rebuke and correction to the Pharisee by firstly placing the situation outside the Pharisee. It's the same thing Nathan did for David, helping him to realize his sin (2 Samuel 12:1-15). How quick we are to see flaws in other people!
7:47 What an amazing thought: forgiveness is the thing that produces love. Love is a fruit of forgiveness. Connecting this back to the healing of the centurion's servant, we might put it this way: worthiness is produced by salvation (this is a way that Luke often writes, with the language of salvation including healing and forgiveness), not the other way around. The woman who was forgiven much is showing that she loves much because of that. In contrast, the Pharisees are rejecting the notion that they need forgiveness, and so they're also not bearing the fruit that comes with it, showing disdain for this Sinner rather than love.
7:48-50 For someone other than God to pronounce forgiveness is to blaspheme. This question that Luke records isn't just a statement of amazement; it's them wondering how this man can presume to blaspheme, which was punishable by stoning. Notice that they don't accuse Jesus of this but discuss it among themselves.
7:50 Faith has been described as "the hand that grasps salvation". This is a good way to look at it. Faith is a gift of God by which people believe His promises and receive His grace. Both this woman and the centurion's servant were saved not because of worthiness, but because of faith.