In this chapter, the author of Joshua tells us the borders and cities assigned to Judah.
I like to think of chapters like these as an archaeologist's dream and a layman's nightmare. This chapter contains a gigantic list of cities, many of which are only ever mentioned here. It is a rich source of information for anyone trying to launch an expedition into the Negev desert, but significantly less exciting to people reading this for religious or theological purposes.
Before mentioning anything else, I should say that this is another "Judah first" chapter, which shows the pre-eminence of Judah over the other tribes. Sometimes lists of the tribes start with Reuben (listing them by chronological age), but many other lists (like this one) start with Judah.
This chapter begins by describing the borders of Judah as it relates to landmarks in the region, and many of these cities have been forgotten or lost. A precise demarcation of the border as it once existed is no longer possible, but researchers have been able to get reasonably good estimates. You can probably find a good map online of the borders of the twelve tribes as they are described here in Joshua. As I think I already mentioned, these borders were never really "real" because Israel never controlled all the land that was promised, and that Joshua assigned as inheritances in this book. The maps you want to look for are "the territories of Israel that Joshua assigned, but they didn't actually control". There are other maps that mark out the land that Israel controlled in various epochs, but don't confuse those ones for this one.
That said, this chapter is a good example of how Israel didn't control all the land they claimed, because v. 12 says the western border was "the great sea", i.e. the Mediterranean, but Joshua 13 points out that the coastline was still controlled by the five Philistine kingdoms.
While I think looking at territorial maps are easier to understand and get a reference for, I will still try to explain roughly where Judah is. Their southern border goes from the southern end of the Dead Sea, through the Negev, and ends near Gaza on the Mediterranean Sea. The border on the eastern side is the Dead Sea up to the mouth of the Jordan river, and then it cuts west (as the northern border) through various meandering turns until it ends at the Mediterranean. The southern end borders on Edom, which is a nation descending from Esau (the brother of Jacob), and if my readers recall the wanderings of Israel, in the book of Numbers the nation passed around Edom because Edom refused them passage through to the promised land. So now Judah is bordering on Edom, having walked around Edom and crossed into the promised land from the East (the former territory of the Amorites).
Verses 15-19 contain a peculiar story about Caleb giving his daughter in marriage to some man who conquered a city for him that was within his inheritance, near Hebron. I'm sure other teachers have extracted all sorts of wonderful metaphors and proverbial lessons from this story, but I don't have much to say about it. Achsah's husband convinces her to ask Caleb for springs of water since her husband's land was in the Negev (i.e. desert). Water is obviously valuable in this arid region. I don't know what else there is to say. Achsah is never mentioned again (except in various repetitions of this story), but interestingly Othniel is actually Caleb's "younger brother" (Judges 1:13, also v. 17 of this chapter), so in this instance Othniel ends up marrying his niece. Awkward, but not entirely uncommon for the OT (cf. Abraham and Sarah). Othniel does appear later, in the book of Judges.
The rest of this chapter is a long list of cities. Many of these cities are referenced in other parts of the bible, but many of them are not. There's all kinds of things I could say about them, but not much of it is immediately relevant to this chapter. This chapter is just trying to lay out the various territories of Judah.
Some of the cities mentioned have not yet been conquered, such as Libnah (v. 42), Ekron (v. 45) or Ashdod (v. 47), so these are aspirational possessions of Judah, if you will.
Lastly, we are told in v. 63 that the Jebusites who lived near Jerusalem could not be driven out, so they lived in the midst of Israel even after Jerusalem itself was captured. This is a striking admission, because it means that even when Israel didn't make a covenant of peace (such as with the Gibeonites) they were not always able to defeat and drive out their enemies. This is in addition to the Philistines whom Judah never conquered at all. Since Israel did conquer Jerusalem itself, I wonder where and how the Jebusites survived. It seems probable that the Jebusites will become a "snare" that the LORD warned Israel about when commanding them to destroy all the inhabitants of the land (Deut 20:17-18). This verse is the first of many similar remarks, because while Israel destroyed the bulk of the native Canaanites, apparently fragments of these nations persisted in the midst of even the stronger tribes of Israel.