This is another one of the more famous stories from Solomon’s life, but taken objectively, I don't believe it is a substantive event. In this episode, the queen of Sheba hears about Solomon and travels to Jerusalem to see for herself. She brings a large caravan of jewelry and spices, and gives those to Solomon in exchange for “all she desired and asked for… out of his royal bounty.” The phrasing is ambiguous, but I think it’s likely that Solomon gave her gifts in return.
I tried to figure out where Sheba is located and got several answers. Wikipedia asserts that Sheba is located in the south of modern-day Saudi Arabia, while traditionally she is thought to be a ruler of some kingdom in Egypt or Ethiopia. Regardless of the real answer, I think it is pretty clear that Sheba must have been a relatively minor kingdom because it did not leave much of an archaeological remnant.
Now, on the face of things this story is relatively straightforward; the queen comes and asks Solomon many “hard questions” about life, the universe and everything, and Solomon astounds her with profound answers, as well as the wealth and splendor that he had accumulated. She gives him several tons of gold, gems and spices, and then departs. I think the story is famous not because of what is in the text, but because of what is not: it is widely speculated that what the queen of Sheba “desired and asked for” was to bear Solomon a son. Obviously the poets elaborate considerably, but the basic notion is that the queen of Sheba was pregnant when she departed back to her homeland.
Indeed, in the previous century there was a large Ethiopian Jewish community (called Beta Israel) that claimed their ancestry through the queen of Sheba and king Solomon. While this origin is impossible to verify, they were granted legal Jewish status in Israel which meant that they had the right of “aliyah”, permission to return to Israel.
The story itself of how they actually got to Israel is messy and the journey was perilous; even after arriving safely, black Jews in Israel claim that they are discriminated against by their fellow white Jews. But somehow, and some way, not only did a large black Jewish community arise, but it also returned to Israel. They claim their origin in the queen of Sheba, which is perhaps more of a poetic statement than a reality. But true or false, this small story played a surprisingly large role in shaping the identity of the Beta Israel community.
Besides that particular controversy, I don’t think there’s really much else to say about their encounter. It's famous, but not a lot happens.
The rest of this chapter is similar to the previous chapter; it contains many details about the finances and administration of Solomon’s rule. He earned 666 talents of gold each year, which is a fabulous amount of money.
Apparently Solomon is earning revenue from “Arabian kings” for some reason? My guess is that this refers to trade moving through Israelite territory as it traverses between the Arabian Peninsula, Africa and Asia Minor. Israel is situated strategically at the crossroads of several major powers (the Egyptians to the southwest, the Assyrians, Phoenicians and Greeks to the north, and Arabians to the east). Verse 29 gives us a brief picture of this when it talks about Solomon importing horses and chariots from Egypt and then exporting them to the Hittites and Arameans. As long as Israel can maintain its independence, it can levy taxes on traffic through the region. Israel's strategic positioning leaves them at risk of foreign aggression, however. They have already been oppressed by their neighbors on many occasions, but in the future we are going to see them fight against stronger, emerging international powers such as the Egyptians and Hittites, the same nations they are trading with at the present moment.
As with the previous chapter, Solomon has horses and chariots located in "chariot cities" as well as "with the king in Jerusalem". This emphasizes the use of chariots for messengers, probably part of Solomon's administrative network.
This chapter also mentions that Solomon is importing apes and peacocks along with gold, silver and ivory (v. 22). I find this a little interesting because it shows that Solomon was interested in exotic animals