In this chapter the daughters of Zelophehad are granted their father's inheritance and Joshua is appointed to succeed Moses.
The first half of this chapter, concerning the daughters of Zelophehad, is a challenge to the culture of male inheritance. It is difficult to nail down exactly what their traditions entail, but we can observe a few things from the text so far.
1) In the life of Jacob, only his sons had an inheritance. To wit, we know that he had a daughter, Dinah, while his entire inheritance was given to his twelve sons. So it appears that a man's daughters inherit his estate only if that man has no sons, which is what v. 8 says.
2) The eldest son is given a double portion of the estate, which is otherwise distributed equally between the sons of all ages. In the case of Jacob, the double portion was taken from Reuben and given to Joseph because of the former's indiscretion and the latter's exemplary behavior.
3) The promised land is given to the sons of Jacob as a permanent inheritance. This is modeled after the inheritance structure normally passed down from father to son, but with a religious undertone because it's part of the covenant between God and Abraham. Possessing the land, and passing it down to one's sons, is therefore both an economic and spiritual act, just as the covenant with God is a spiritual covenant that guarantees material blessing (for instance, Lev 26:1-13 which highlights both the material and spiritual blessings of following the LORD's covenant).
4) Women are not given their father's inheritance because when they marry, they become part of their husband's family. It's not that women don't get an inheritance, it's that they don't get one from the family of their birth; they receive an inheritance from their husband's family through marriage. Furthermore, because women become part of their husband's family, all of their possessions become part of their husband's wealth. We see a vague reference to this in Gen 31 when Rachel and Leah say that "Do we still have any portion or inheritance in our father's house? ... All the wealth which God has taken away from our father belongs to us and our children".
In context, remember that Jacob had taken Laban's wealth by shepherding his flock and receiving all the speckled or striped animals. Also remember Laban had no sons, so we can guess his inheritance would go to his daughters for that reason. However, his daughters say that he "entirely consumed" his wealth and deprived them of their proper inheritance. Therefore Jacob was justified in taking Laban's wealth. The inference is that the inheritance of Rachel and Leah is given to Jacob.
This third point is at the heart of the complaint lodged by Mahlah et al: "Why should the name of our father be withdrawn from among his family because he had no son?" (v. 4). The essence of this complaint is that the promised land is a permanent inheritance for both the tribes of Israel and the individual families and men in the tribe. What the daughters of Zelophehad are saying is that if their father's inheritance were given to e.g. his brothers or the tribe at large, then his "name", i.e. his estate or the family of people descending from him, would be wiped out and therefore he would not have a permanent inheritance anymore.
Maintaining a possession in the promised land is part of the covenant, so giving Zelophehad's inheritance to his daughters is necessary to help maintain the covenant to Zelophehad. Keep in mind that they are primarily asking about the distribution of land and not Zelophehad's inheritance of animals, gold, etc., because those are a temporary inheritance while the land is meant as a permanent inheritance as I've discussed on many prior occasions. Verse 7: "You shall surely give them a hereditary possession among their father's brothers".
In summary, the covenant of Abraham is a covenant to both the nation but also to the individuals within that nation, so the LORD is taking steps to protect their inheritance at large as well as the inheritance to a single man, Zelophehad.
In response, the LORD grants their request and specifies the order of inheritance in case a man has no sons. This order of inheritance is probably very similar to what they were already practicing, except that daughters can now inherit a man's estate before it goes to his brothers, etc.
The second half of this chapter tells us that Moses will die before entering the promised land, as we already knew. The LORD appoints Joshua to follow Moses, which should not surprise us either. He has a long and distinguished history by this point, having commanded the military against the Amalekites (Ex 17), served Moses as his attendant (Ex 24:13, Ex 33:11, Num 11:28), and was one of the two faithful spies (Num 14:6 et al.).
All of these factors combine to make Joshua an excellent candidate for succeeding Moses as the leader of Israel. His military training will enable him to lead Israel into battle. He is also one of the only two surviving Israelites to see the promised land, giving him valuable insight for the coming battles. His service to Moses gives him experience needed to administrate the nation as a judge. His experience dwelling in the tent of the LORD (Ex 33:11) means he can interact with the LORD and govern the nation spiritually as well.
Also, Joshua is "a man in whom is the spirit" (v. 18), so he should prove to be an excellent leader indeed. So Joshua is commissioned and should be ready to lead the nation when they enter the warfare of the promised land itself. That story, however, does not begin until we reach the book of Joshua. For now, Moses will remain the dominant figure in Israel and will remain so for the rest of the Pentateuch until his death in Deuteronomy 34 at the very end of the book.