Most ancient religions and philosophies have a secret "esoteric" doctrine revealed to a select number of initiates, as well as an "exoteric" doctrine that is revealed to the common people. When it comes to Buddhism, the esoteric and exoteric principles are very similar, though there are some details of esoteric Buddhism that you won't see in most Buddhist texts, for example, what is known as the Septenary Constitution of Man, or the seven principles that constitute a human being. I read about this in the book Esoteric Buddhism by the Theosophist A. P. Sinnett. One particularly interesting quote is:
From A. P. Sinnett's Esoteric Buddhism (1883):
"All things, not man alone, but every animal, plant, and mineral have their seven principles, and the highest principles of all--the seventh itself--vitalizes that continuous thread of life which runs all though evolution, uniting into a definite succession, the almost innumerable incarnations of that one life which constitute a complete series."
So first of all, what are the seven principles? Although they can be listed out as distinct principles, they are all intertwined, so bear in mind that listing them out fails to capture the connections between them. I'll call them by their English names and give the Hindu words in brackets:
1) The Body (Rupa)
2) Vitality (Prana or Jiva)
3) Astral Body (Linga Sharira)
4) Animal Soul (Kama Rupa)
5) Human Soul (Manas)
6) Spiritual Soul (Buddhi)
7) Spirit (Atma)
The first principle is simple: it's just the body, containing our organs, the molecules out of which they're formed, all the way down to the basic constituents of matter (mostly protons, neutrons, and electrons). On this level, everything is material: it is the matter that forms a human, the same matter that forms all physical objects such as plants, human or animal bodies, ice-cream, mountains, the air.
The second principle, vitality, is the start of life. It is organic matter. Like the body, it is still physical: it is the energy or force of a human. If we think of matter (m) and energy (E) as interchangeable (thank you, Einstein: E = mc^2), then the principle that vitalizes the body isn't distinct from the body, and is material as well. When the body of the living creature dies, the vitality is no longer united in the body but dissipates among the particles out of which the body was formed. This is because the higher principles that "hold the human being together" leave the body to reincarnate elsewhere, whereas the vitality doesn't undergo reincarnation.
The first, second, and third principles all perish when a human being dies, though the higher principles, which we will look at shortly, are reincarnated in this world or another, continuing our spiritual life.
The fourth principle is the animal soul, also known as the vehicle of will. It isn't developed in living beings like plants, but only in animals (and hence, us). It includes emotions, passions, instincts, and sensations. It is focused on the material world and how we can gain pleasure and avoid pain. It is the instinctual part of our mind, the "id" if we're using Freudian terms. There is no intellectual thinking here, just instinct born of the natural world we live in.
The fifth principle, the human soul, involves thinking and individual consciousness. This is where our reason and memory lie. Those who are more enlightened have a more fully developed fifth principle. Most people are unable to connect with this part of themselves, or if they are, then it is rather as a slave to their fourth principle than as the rightful master over it. Yet just as individual people can be more fully evolved than others during the same time period, so too does the human species as a whole develop by evolution, and so as a whole, our species is still in the process of developing our fifth principle.
The sixth principle, or spiritual soul, is our spiritual consciousness that connects us to everything in the world, as well as forming the source of our morality and compassion. It is our individual soul, our "higher ego". Since our fifth principle is undeveloped, it goes without saying that our sixth principle is as well. The spiritual soul is the vehicle for the seventh principle, so can be seen as a more individualized spirit, because as we go down to lower principles, we become more individualized and more material.
The seventh principle is the spirit itself. It is undifferentiated, and in essence, is the same spirit in everything. It could also be considered to be God, Buddha, or the One, the most abstracted part of us. We cannot describe it, for to describe it as anything but simple and pure would be to assign characteristics to it, and as it has no counterpart in the physical world, we will always fall short in our descriptions.
There is much more that can be said about each of these principles, but this is the general idea.
Now, the quote above touches upon how the principles are linked. The seventh principle, the universal spirit, is what binds the principles within us. What makes all these principles constitute a human being (or whatever other creature your higher principles are incarnated in) is the seventh principle, because the lower principles can be seen as particular instances of the universal spirit. The higher principles are bound together through multiple incarnations, though the three lower ones (body, vitality, astral body) are different for particular lives.
To use a common analogy from many occult traditions, it is like white light being dispersed by a glass prism: although it is in essence pure white light, it can be split into many colours. Likewise, the universal spirit can be "dispersed" into many spiritual souls, which can then be further dispersed into human souls and so on. Although things like plants and minerals don't have their human and spiritual souls developed, these principles are still present, only in a dormant form.
This also provides the answer (at least, part of the answer) to the age-old question of how something immaterial (e.g., the soul) can interact with something material (e.g., the body). It is because they are all made of the same "stuff", but just in different proportions, with matter at one end of the spectrum and spirit on the other. This is what Sinnett referred to as matter existing in other states that we can't perceive by the five senses. And since the same kinds of thing can interact with themselves, as we see in the physical world where matter interacts by the four forces (gravity, electromagnetism, strong nuclear force, weak nuclear force), the different principles can also interact with one another.