In this chapter, Moses gives us an assortment of various laws.
Like the previous chapter, I don't see any overarching themes in this chapter. Rather, it appears to be a grab bag of miscellaneous laws. These chapters are always the hardest for me to meaningfully commentate because there isn't any one thing that I can point to and say, "this is what the chapter is about". Since I can't do that, all I can do is address each law individually and try to show how it fits into the cultural context of the day.
Verses 1-3 establish that a man can only be beaten with 40 lashes. This is interesting because we haven't been given a single law that orders punishment by flogging. We know that the judges would have broad discretion in punishment for lesser crimes (crimes not specified by the law of Moses), so it's probable that's what these verses refer to. Anyway, in Jewish history the command to not lash more than 40 times was modified to not lash someone more than 39 times (sometimes stated as 40 minus 1 as in 2 Cor 11:24), to add a margin of safety in case the punishers miscount while lashing someone.
Verses 5-10 establish a law of Levirate marriage, as was previously mentioned in the story of Er, Onan and Shelah, the sons of Judah (Gen 38). In the time of Judah, this was just a cultural tradition, but now it is also encapsulated in the law of Moses as binding upon the house of Israel. To the best of my knowledge, this law is never referenced again in the rest of the bible, which possibly shows it fading out of significance as Israelite culture shifted over the centuries.
Verses 11-12: I have no words to describe this law. Let this be a warning to all you genital grabbers out there: the judges will "not show pity" to you.
Verses 13-16: This is another basic "justice" law, which treats justice as not just a matter of upholding the law equitably, but also equitable business dealings. A bit of background might be useful here: in modern life, most measuring scales use spring tension to estimate weight, which means that they don't require a counterweight. It is using the mechanical resistance of a spring under pressure to figure out how much a given thing weighs. In ancient times, the scales were simply levers used to weigh one thing against another, and when the two things weigh the same, then the scales "level off". What this means is that in order to know that something weighs one shekel (or pound or kilogram or...) you need to have a weight that weighs that same amount. An unscrupulous merchant can "cheat" the scales by having a set of weights that are too heavy or too light, using the light weight when measuring out the grain (or whatever) you are buying, and the heavy weight when measuring out the silver you are paying him. By doing this, he can give you less grain for more money and you are none the wiser of his scam.
This law is about using equitable weights, because to use otherwise implies that you are scamming whoever you are making a business deal with. I wrote about this briefly when Abraham bought the field of Machpelah using the "merchant's shekel" (Gen 23:16), because the merchant's weight is likely to be manipulated in order to cheat Abraham (the foreigner).
Verses 17-19 conclude the chapter by reminding us that the Israelites are to destroy Amalek (Ex 17, Num 24:20). In case the people forgot or something, I don't know.