In this chapter, the LORD tells us the laws of the Nazirite, concluding with the formula for the priestly benediction.
The Nazirite vow is very interesting. As we can see in this chapter, it is a voluntary vow to the LORD, possibly one of several kinds of vows, but the only one clearly defined in the OT. It is a vow over a certain time period, but we know from some biblical examples that the time frame does not have to be narrowly defined.
There are three well-known Nazirites in the bible, who all maintain the vow throughout their entire lives: Samson, Samuel and John the Baptist. In all three cases they were called to be Nazirites before they were born, so it's a bit different than the pattern in this chapter. This chapter seems to suggest that the Nazirite vow is something undertaken by a person for some period of time, which is symbolized by the growth of that person's hair over the duration of the vow. While the biblical examples are all men, verse 2 makes it clear that this vow is for both men and women.
There are three main clauses to the Nazirite vow: avoidance of wine, growing out one's hair and avoidance of dead bodies under any circumstances. This is a pretty strange list, and at first glance there is no apparent pattern to it. Avoiding dead bodies is perhaps related to the Levitical regulation, whereby approaching a dead body results in ceremonial impurity. Similarly, priests are prohibited from approaching dead bodies under any circumstances, except their close family. So perhaps we can see a connection there, where the Nazirite vow is like taking up a measure of the sanctity that is expected of the priests. The Nazirite vow is clearly harder than that of normal priests and is instead comparable with the restrictions on the high priest, who cannot approach any dead body, even that of his mother or father.
What's peculiar about this is that, just as with the priests, it is not a prohibition on ceremonial impurity, because there are many ways to become impure and approaching a dead body is just one of those ways. There is no similar restriction on e.g. touching an unclean person, being spit upon or touching other bodily fluid, or any of the other various sources of ritual impurity.
Additionally, the prohibitions on wine and shaving are both new and I don't believe there is any meaningful parallel in the Mosaic laws we have read so far. There was a restriction on shaving the corner's of one's beard (Lev 19:27, Lev 21:5), but that was part of the prohibition on adopting the religious customs of Canaan. In this case, the Nazirite is not shaving his or her head as an affirmative act, whereby the hair on the Nazirite's head is intended to be an outward sign of consecration and the vow itself. It is an externally visible action to mark the person as a Nazirite to anyone who encounters the Nazirite. Other than that, I don't know the significance of growing out one's hair.
The prohibition on drinking wine is unexpected, but I can think of two possible meanings for it. The first is that it's a sign of mourning, because drinking wine is usually associated with joy or celebration. The second is that the Nazirite is supposed to be sober and to avoid the intoxicating effects of the wine. This is probably related to the first meaning, because soberness is usually related to prayer or mourning. I.e. the Nazirite vow is intended to be sober and a dedication to prayer. Prayer itself is frequently associated with pleas for mercy or forgiveness of sin.
From these prohibitions we can see the Nazirite vow is exceedingly strict. The prohibition of wine specifically includes vinegar, grapes, grape juice, raisins and even the seeds and skins of grapes. It is obviously meant to be a very thorough restriction, and though it is not very broad (in the sense that it's only banning grape products), it is very deep (in that it bans literally all of them). Similarly, the prohibition on approaching dead bodies is extremely strict, such that it has a ritual in case someone unexpectedly dies near you. In that case you have to make an offering and then restart the vow from the very beginning. Clearly this vow is not meant to be taken lightly. Even so, these restrictions do not seem like they would impede on normal life, in the sense that very few people have to drink wine or shave.
Also, the cost of the offerings when you complete the vow is relatively high, since it requires the normal course of a burnt offering, sin offering, grain offering and peace offering. Unlike many other rituals, this one does not have an scaling factor for economic hardship, probably because it is a voluntary offering so if you can't afford it, you can simply refrain from taking it.
To be honest, I don't think the details of the vow are as important as the bigger picture, which is that this vow makes a pattern for any person to consecrate him or herself to the LORD. After having spent so much time learning about the priesthood and the rituals of the tabernacle, it is natural to feel like the Levites and sons of Aaron are monopolizing access to the LORD. What's worse is that those roles are hereditary, so a person born into another tribe will never be allowed into the priesthood or Levitical service. However, the Nazirite vow is a pattern for any person to volunteer that same sort of consecration we see demanded of the priests. As I pointed out above, avoidance of dead bodies is one of the restrictions on both priests and the Nazirite.
This consecration comes with a great deal of restrictions. In Leviticus we saw Nadab and Abihu literally consumed by fire from the LORD's presence when they performed a "strange" (i.e. unknown or unauthorized) offering of fire, so there should be no doubt that the priesthood is a very narrow road to tread. In the same vein, the Nazirite vow is exceedingly strict because that is what holiness of God demands: holiness itself requires obedience, narrow obedience, and adherence to the commands of the LORD. The priests have both a blessing of nearness to the LORD's presence and a curse of death if they fail to keep any of the LORD's commands. The strictness of the law is not nearly as great for laypeople, but laypeople do not share in the commensurate blessings of entering the tabernacle and seeing or touching the holy things.
The vow of the Nazirite is therefore a path for anyone to enter into a similar form of consecration. The requirements (avoiding wine, shaving) are different, and Nazirites are not given access to the tabernacle or the other regulations of the priesthood, but they share the sense of consecration and devotion to the LORD. So that's why I really like the Nazirite vow: it is democratic in a way that the priesthood is not. The priests do not have any choice as to their role, but the Nazirites can volunteer their consecration.
This chapter concludes with the priestly blessing, which is short and simple but still meaningful. Among other things, it shows us what sorts of characteristics the Hebrews valued. The most significant blessing is peace, "shalom", which concludes the benediction. There is a three-fold repetition of the divine name, Yahweh, which is probably why v. 27 talks about "[invoking] my name on the sons of Israel". The priest is figuratively "calling upon the name of the LORD" to bring a blessing to the people.
The other significant part of the blessing is that it twice invokes the face of the LORD, first by referencing his face, "penim", shining upon them (like the sun, or a candle). The second is a reference to his face, "penim", being lifted up or raised towards them.
Structurally, there are three separate blessings that all begin with the divine name and have two expressed blessings. The first triplet is "bless you", "face shine upon you", "lift up his face towards you", the second triplet is "keep you" (i.e. hedge or guard), "give you grace" (i.e. favor or esteem), "give you peace".
Peace, or shalom, is probably the most complex of the blessings because the word is widely used and it has many subtle implications. A quick search of the OT finds 236 separate references to shalom starting in Gen 15:15 and ending in Malachi 2:6. The usage of shalom here is probably one of the most subtle, because it's not referring to peace with respect to some concrete event, circumstance or relationship. It is a general and deep tranquility, perhaps opposite of the kind of strife and labor that began in Gen 3:17, "in toil you will eat of [the ground]". Instead of the toil, sweat and sorrow of the curse of Adam, the priestly blessing brings protection, favor and inner tranquility and peace to the recipient. All of these blessings are strongly associated with the LORD and with his presence (panim).
Really though, I think the Aaronic blessing is really more easily learned by experience than by analysis. So I would encourage to simply pronounce the blessing over someone, whether another person or yourself. Try to feel the words as you speak them and meditate on the imagery these words evoke. Unlike many other parts of the OT, this is a part that anyone can utilize today, so I think it's an opportunity worth taking.