A. General Remarks on Hakafos.
We engage in hakafos everyday of Sukkos except Shabbos and on Simchas Torah.
On Hoshana Rabba, we do 7 hakafos and we do them again (twice) on Simchas Torah.
The general idea of Hakafos is that in some spiritual way, we do not fully perceive, it breaks down barriers just as encirclement caused the walls of Yericho to collapse so the hakafos break down the walls of separation that divide us from Hashem.
(This is also why a kallah walks around a chossen to break down the barriers and emotional distance that may exist between them).
Rav Yosef Ber Soloveitchik z"l explains that during Sukkos we encircle the Sefer Torah by walking around with a lulav essentially declaring that through mitzvos that express our obedience to Hashem, we will be able to understand the Torah.
But then on Simchas Torah, we encircle the Bimah which is empty by carrying a Torah.
If Stage 1 says we connect to Torah via Mitzvos, Stage 2 says we connect to God through studying the Torah which is God's mind and essence.
Thus, Gra says the Bima is mamesh the place of the Shechina and he did not allow children to sit on the platform.
Let me also mention the heara of Chidushei HaRim who said when we hold a Sefer Torah and dance, we are making an oath by holding the holiest of objects that we live our lives in accordance with the Torah.
One has to think about this; hakafos are a big deal.
In any case, each hakafa releases a discrete and unique spiritual energy and one should try at least at the beginning or end of the hakafa to have kavanna for that energy so that they can become a kli to receive it.
May we be zoche to feel that connection!
R. Yeruchem of Mir used to say that he doesn't know if Hashem gets more nachas from our neila or from our hakafos.
B. Specific Comments on Individual Hakafos
Sefiros are bold.
First Hakafa = Avraham = the midda of loving kindness and Chessed .We ask Hashem to give US the quality of being kinder ,more compassionate people.
Second Hakafa = Yitzchak = the midda of Gevurah, inner strength and discipline. We ask Hashem to make us stronger, more focused and intense in doing what is right and avoiding sin.
Third Hakafa = Yaakov = the midda of Tiferes, beauty, synthesis, balance,integration, harmony. Chessed carried to excess can be destructive ,eg ,in raising kids. Thus, Avraham's chessed produced Yishmael. Too much gevurah can also destroy. Yitzchak produces an Eisav .Only when chessed and gevurah are properly mixed do you get a perfection and a harmony. Yaakov represents the blending. Thus, all of his children are holy and comprise Am Yisrael. So after we prayed that we should have chessed and gevurah, we ask Hashem for the capacity to balance and mix them in the right proportions.
Fourth Hakafa = Moshe = Netzach = eternity, victory, perseverance. That is why Ein Adir, a song in honor of Moshe, is bdavka sung in the fifth hakafa .T uhe quality of Netzach is the ability not to be defeated by the external circumstances in your life. Even if a person has the proper blending of chessed and gevurah which is tiferes, he will be unable to carry out his mission because of an unsupportive family or community or job difficulties or health. So Netzach is the quality that can make me unstoppable inspite of these impediments. This was the quality of Moshe who had a mission and carried it out despite opposition, criticism and rebellion from millions of people.
Fifth Hakafa = Aaron = Hod = glory, opposite of Netzach. Netzach is the capacity to ignore people/distractions Netzach is necessary to accomplish goals but it can also mean that you have no considerations for the needs of other people .You are like a steamroller. Aaron was the ohev shalom, rodef shalom ,who saw the good in every person and let them shine. So after we ask Hashem for the quality of endurance and perseverance, we ask for the opposite quality of being able to step back and pay attention to other people, This might be the reason why this hakafa as Ozer Dalim is a segula for parnassa; when we open our heart to others, Hashem opens Himself to us to give us what we need.
Sixth Hakafa = Yosef (Note that al pi kabbalah, Yosef comes after Moshe and Aaron although he is chronologically earlier because he represents the synthesis/combination of their two middos This is why the Arizal's order of Ushpizin puts Yosef on Day 6) = Yesod The same way the third sefira/hakafa of Tiferes is the balance and integration between chessed and gevurah, the sixth sefira of Yesod is the balance and integration between the aggressiveness of Netzach and the passivity of Hod. Too much netzach, you are a steamroller; too much hod, you are a dishrag who is so busy paying attention to others that you never formulate and accomplish your goals and mission. Too much hod, you build an eigel because who are you to oppose the people's wishes? Yosef himself was able to achieve this balance. On one hand, he had the inner strength and fortitude not to be crushed or influenced by the Egyptian culture whether in poverty or affluence. He remained impervious to all the negative influences that could have destroyed him. But at the same time, he was able to manage and supervise a world economy by making others look good and revelling in their success, a prime attribute of a successful manager. (R. Moshe Sherer z"l used to say that the literal meaning of the phrase that Yosef was an ISH MATZLIACH was not that HE was successful, rather he caused OTHERS to have success and that is why he indeed was successful. Good business advice!) So after we ask Hashem for the qualities of Netzach and Hod, we ask Hashem for the power to balance them and use them in good productive ways.
Seventh Hakafa = Dovid (that is why Dovid Melech Yisrael Chai Vkayam belongs to the seventh hakafa, not earlier) = Malchus The first six sefiros ar two triads in which opposite forces are balanced and integrated. According to the Zohar, the seventh sefira of Malchus is an emty vessel; it is simply a receptacle for all the spiritual powers that preceded it. That is why Malchus is also referred to as Nukva - the feminine element that receives the seed of the male. Dovid HaMelech understood that he had nothing on his own; everything was a gift from Hashem. Even his very life was not his own ;he got 70 years from Adom Harishon. His life was nothing more than a reflection of the glory of God. This is also the connection between Dovid and the moon which also does not have light of its own but simply reflects a greater light. Thus, in the last hakafa, we pray that our lives may be devoted totally to reflect the glory of Hashem, to be a conduit for HIS light rather than for our egos.