In this chapter, the genealogy continues with the descendants of Jacob and Judah.
By and large, I don't expect my readers to know or care about most of the people named in this chapter. I don't expect it because I don't know or care about most of them myself. For instance, you don't really need to know a whole lot about Jerahmeel, son of Hezron, to understand even the book of Chronicles, much less the bible as a whole. Rather than focus on the names and people, I'm going to focus on the structure and layout of the genealogy to see which family lines are being emphasized. I will point out a few individuals that have names that interest me, however.
First of all, the genealogy lists all twelve original sons of Jacob (the original twelve with Joseph and Levi, not the "updated" twelve with Manasseh and Ephraim). However, out of the twelve it only lists the descendants of Judah. Although David is relevant to the whole nation of Israel, I think this shows a unique emphasis on Judah and the southern kingdom vs. the northern kingdom, which perfectly fits the southern context of this book. The northern kingdom had already been destroyed and swept away before Chronicles was written, which means that in addition to the southern context of Chronicles, many of the genealogical records from the north were probably also destroyed, making an extensive northern genealogy impossible to recreate.
Secondly, David was the 8th son of Jesse (1 Samuel 16:10-11), but in v. 15 David is listed as the seventh son. Although it's hard to say for sure, this discrepancy likely exists because the author of Chronicles wanted David to be the seventh son because seven is the number of completion or fullness. Since David represents the "completion" of God's authority and kingdom on earth, David could perhaps be thought of as the seventh son. Even though the firstborn is usually more highly favored, it would have been well-understood from their historical narrative in Samuel that David was not the firstborn, and indeed being the youngest and the weakest and smallest is part of David's kingship story, it's part of God's general plan for honoring the younger over the older (cf. Jacob over Esau, Isaac over Ishmael, Judah over Reuben, Perez over Zerah, etc.) But changing David from the 8th son to the 7th son is hardly noticeable to most people and it puts slightly more symbolic weight behind him. This is just speculation on my part, but most commentaries I've read suggest that Samuel and Kings are close to a historical account, while Chronicles is more of an idealized history that subtly reshapes certain figures, particularly David and Solomon, to symbolize their hopes for a Messiah.
Also, when it comes to David's genealogy I think it's interesting that Ruth is not mentioned at all. Actually, Ruth is not mentioned anywhere in the bible at all except for the book of Ruth and a single reference in Matthew 1. It's possible this is because Ruth is a woman and most women are omitted from biblical genealogies, but at the same time it's also possible that Ruth is omitted because the book of Ruth may not have been available to the author of Chronicles or it may not have been broadly distributed in this historical epoch. Given that the author of Chronicles seems to have a copy of nearly every other part of the Deuteronomistic history, and given that there must have been some copies of Ruth for it to have survived into the modern era, I think it's most likely that there were copies of Ruth floating around but the Chronicler did not consider them authoritative scripture or relevant to his intended audience.
On a similar topic, verses 16-17 mention a sister of David named Abigail. I think it's worth pointing out that this is a different woman than the Abigail from 1 Samuel who David married.
A few other names worth pointing out:
Bezalel constructed many of the furnishings of the original tabernacle made by Moses. My NIV study bible suggests that the author of Chronicles is trying to draw parallels between Bezalel and Solomon, essentially saying that Solomon constructs the new temple the way that Bezalel built many components of the older tabernacle. I don't think the connection is all that obvious, and I'm also not sure why it's important, but it does seem plausible to me that this was done on purpose.
Ephrath is another name for Bethlehem, but in v. 19 it is also the name of a wife of Caleb. As far as I know there is no connection between the name of the town and the person, they just happen to be the same.
In terms of the genealogy structure, this entire chapter forms a chiasm around the three sons of Hezron: Ram, Caleb and Jerahmeel, in that order. The chiasm (ABCCBA) is centered on v. 33, where the phrase "these were the sons of Jerahmeel" indicates the end of Jerahmeel's first section. Since it continues talking about the descendants of Jerahmeel, this indicates the second section about Jerahmeel in the aforementioned chiasm. Caleb's second section begins in v. 42, and the chiasm concludes in chapter 3 with the descendants of David (who is himself a descendant of Ram). Coincidentally, one of the sons of Jerahmeel is also named Ram, but this is a different person from the Ram son of Hezron. Because of the way the chiasm is structured, it mentions David both at the beginning of the chiasm (v. 14) and at the very end (chapter 3). The author of Chronicles focuses a considerable portion of this genealogy on David, which forms the background for the later historical section.
Interestingly, much of the section on David's descendants may not have been a part of the original genealogy. I say this because of the parallelism between v. 50 and 1 Chron 4:4, where v. 50 begins a section on the descendants of Hur and 4:4 ends that section. This most likely implies that there was some original genealogy that lists the descendants of Jerahmeel, Caleb and Hur, and the author of Chronicles inserted some other genealogical fragment into chapter 3 relating to the descendants of David. I think this is interesting because it shows how the Chronicler is shaping the genealogy to suit his purposes.
Lastly, there is another extended chiasm around Shelah, Perez and Hezron that extends from 2:3 through 4:23. It begins by mentioning these three men (Shelah and Perez are brothers, Hezron is a son of Perez) in verses 3-9. Then the entire section from 2:9 through 3:24 is entirely dedicated to the descendants of Hezron (which itself contains the previously mentioned chiasm around Ram, Caleb and Jerahmeel). Right when you think the Chronicler forgot about Shelah and Perez, chapter 4 begins with a section on the descendants of Perez and concludes in 4:21-23 with some of the sons of Shelah.
In terms of the overall framing of these chiasms, the outer chiasm of Shelah, Perez and Hezron is designed to give us an overview of the descendants of Judah, while the inner chiasm (the sons of Hezron) is important for framing the genealogy of king David, who is central to the later historical portion in Chronicles.
The most striking omission from this overall structure is Zerah, the last remaining son of Judah. Even though the descendants of Zerah are mentioned in v. 6-8, Zerah is not part of any of these chiasms, and I'm not entirely sure why. I don't think Zerah is important to Chronicles, but neither is Shelah so I don't know why Shelah is part of the chiasm and Zerah is not.
In conclusion, even though I discussed some of the individual names, I think the most important thing to look at is the overall structure of the genealogy, and from that we can learn what is the purpose of the genealogy and how it fits into the larger structure and themes of Chronicles as a whole.