In this chapter, Moses tells the nation rules concerning vows, particularly relating to women.
I had wanted to write up this chapter yesterday, but I really struggled to keep up with the topical shift. This chapter really has almost nothing to do with the previous chapter, which was about the sacrifices to be made at various festivals. That chapter was written in the style of Leviticus: numerical, procedural and precise; the subject matter was also very priestly, relating specifically to sacrifices and festivals (which would have been hosted at the tabernacle, the center of priestly authority).
This chapter, while still a legal discussion, is much more about social law than religious law (of course, everything here is religious to some extent, but the focus of this chapter is on governing personal behavior not otherwise connected with religious life). This chapter has no numbers, no offerings and no priests involved. This chapter is definitely a statement of law, however, so it has that in common with the previous chapter. Before moving on, I'll just say that this rapid shift between topics is characteristic of the Pentateuch and we have seen it time after time in the chapters and books we have covered so far. I have said on a few occasions that the author "does as he pleases" and if he chooses to rapidly change topics, then that's what will be. But it is worth paying attention to, because the structure of the text reveals the structure of the mind who wrote it.
Moving on. This chapter is ostensibly about vows, but in reality it is really more a discussion of the authority that a father/husband has over the words of his daughter/wife. This isn't the first time we have seen this principle in effect. We have seen several occasions where the wives or daughters of men were punished or rewarded for the deeds of those men. For instance, at one point Lot offered to give his two daughters to an angry mob of townsfolk waiting to rape his angelic visitors (Gen 19). In this situation, his daughters are left without recourse and unnamed. Clearly Lot holds substantial, if not total, control over their lives. Later in the same day, those same angels save Lot, his wife and his whole family from impending doom primarily because of Abraham's intercession in Gen 18. But in that episode, we can also see that Lot's family were saved essentially because of their relation to Lot (who himself was mostly saved because he is the nephew of Abraham), and not because of anything they did or were.
Another example is the rebellion of Dathan and Abiram, whose families perished with them (Num 16). It was not because of anything their wives or sons did that they died, but simply their relations to Dathan and Abiram. This is because Dathan and Abiram, as the fathers of their household, have an essential control over their families both to direct their behavior and to shape their destiny. I'm not saying that this is right or wrong, but it's certainly how the biblical author viewed such matters.
The last example I will give is the biblical commands in Ex 20:12, 21:15 and 21:17. These are the commands to honor your mother and father, and that anyone who strikes or curses their parents shall be put to death. In these cases, it is both the mother and father who must be honored by their children, which establishes a certain hierarchy: the children honor their mothers and fathers, wives are beholden to their husbands, and the father rules the entire clan. For their part, men are expected to submit to their elders within their extended family, so as men grew older they were given greater esteem and power.
What we see here is that this authority extends also into the realm of vows, that a male relative can overrule any vow made by a female dependent as long as they do so upon immediately hearing of the vow for the first time. This is similar to the principle that legal motions must be made in a timely manner. This basically requires you to take some action (in this case, overrule a vow) at the earliest possible instance, rather than wait and overrule it later if it becomes inconvenient. If the man delays, then he loses the right to overrule that vow and it becomes binding.
We also see a transference of male authority as a woman marries, passing out of the household of her father and into the household of her husband. In that case, her husband has a one-time chance to annul any vow that she made before marrying him.
Overall, I think this chapter is relatively straightforward, so I don't think there's anything else I need to add.