In this rich scriptural passage, Elijah stands for Christ. He builds the altar upon the foundation of the twelve tribes (apostles). The wood on the altar represents the wood of the Cross; the bull represents the sacrificial offering Christ made of himself. The four jars of water represent the four Gospels, and the water itself represents the washing of baptism. The fire of the Lord is the consuming fire of God's love, which transforms the very substance of the elements offered on the altar until all is taken up in God. This complete acceptance of the sacrifice, together with the punishment meted out to the prophets of Baal, is a foretaste of the end-times, when Jesus will return to judge each according to his works. The passage foreshadows both the sacraments of baptism and Eucharist and demonstrates the importance of the priesthood. For another passage related to the priesthood, think about the correspondences between what happens to Jesus in his passion, death, and Resurrection and what happens to Peter in Acts 12.
In 1 Samuel 16:1-12, the interpretation revolves around the fact that the word "Bethlehem" means "House of Bread." Samuel is told by God to go to Jesse's house in Bethlehem and anoint a new king. If anyone asks why he is going, Samuel is to tell him that he is offering a sacrifice that involves a banquet meal. Every attendee of the banquet had to be ritually clean. Upon entering the House of Bread, Samuel incorrectly identifies who is to be king. As a result, God warns him not to judge by appearances when looking for the King in the House of Bread but to rely on God, who judges what is in the heart. Seven of Jesse's sons appear, but none are suitable. It is only the eighth son who is suitable. It is useful to recall that Jesus rose one day after the seventh (Sabbath) day, that is, on the eighth day. Thus, when we go to the House of Bread looking for the King, we are not to judge by appearances but rather by what is in the heart -- what is really there. When we do, we will find the Son of the eighth day, and we will know our King. The entire story points to the Eucharist.
Note that 1 Chronicles 11:16-19 and 2 Samuel 23:14-17, which also refer to Bethlehem, have similar Eucharistic overtones. In those passages, water is obtained from the cistern of Bethlehem through the work of three mighty persons. David connects Bethlehem's water with their blood and pours it out on the ground as the priests do the blood libations at the altar.
Proverbs 1:1-6 provides additional confirmation. In this passage, Solomon tells us that he is writing the book of Proverbs in order to teach us how to understand a proverb: That is, he will use proverbs to explain proverbs. Jesus was a wiser man than Solomon (cf. Mt 12:42), and he taught constantly using parables (Matt. 13:34-35). Yet Scripture witnesses again and again how he had to explain every figure he used (e.g., John 10:6). This is important for two reasons.
First, it confirms what both Luke and Matthew were inspired to record about Jesus: "the people who sat in darkness have seen a great light" (Matt. 4:16, Luke 1:79). Many of the people of Christ's time, who lived under the Old Testament dispensation, were incapable of reading Scripture with the necessary New Testament understanding. Their minds needed to be enlightened to understand the parable they and their ancestors daily lived out.
Second, as Matthew 13:10 records, Jesus gave the apostles the ability to understand what the Old Testament "parables" pointed to. Indeed, Jesus promises exactly that in John 16:25 -- there would come a time when he would speak to them plainly and not in any figure. However, he makes this promise just scant hours before the Twelve break and run like water. The promise is fulfilled only after the Resurrection: "Then he opened their minds to understand the scriptures" (Luke 24:45).
Unfortunately, not all new Christians shared this apostolic gift. The Thessalonicans in Acts 17 who rejected apostolic guidance in how to read Scripture continued on in darkness, while the Bereans, who desired the full understanding of Scripture, were unable to attain it without the guidance first of Paul and then of Silas and Timothy.
Scripture itself underscores the extent to which this apostolic guidance is necessary. No person or group of people in Scripture attained an accurate understanding of who Jesus was or what he did without authorized guidance, despite the fact that knowledge of Jesus was widespread ("This thing was not done in a corner" [Acts 26:26]). This apostolic gift explains why the epistles are rich in the four-fold sense of Scripture. The whole letter to the Hebrews, especially passages like 8:4-5, 9:24, and 10:1, refer repeatedly to the Old Testament through the four-fold sense. And though Colossians 2:16-17, Galatians 4:24-30, 2 Peter 2:6, James 5:10, and Jude 7 all demonstrate the concept, 1 Corinthians 10:1-11 shows the principle at its clearest.