Land animals, which were created from the soil, are rendered fit to eat by the severing of both vital passages (the windpipe and the gullet). Fish, which were created from the water, do not require any shechitah to render them fit to eat. Birds, which were created from a mixture of soil and water, are rendered fit to eat with the severing of either one of the two vital passages.
Talmud, Chullin 27b
In the terminology of Kabbalah and Chassidism, soil and water are analogs for materiality and spirituality. Aside from the usual association of soil with earthiness and mundanity, and of water with purity and sublimity, the difference between soil and water expresses one of the basic distinguishing characteristics between matter and spirit. Soil is comprised of distinct granules, while water forms a cohesive expanse. When two types of soil (or any two solids) are combined, they remain separate entities, however thoroughly mixed; liquids, on the other hand, blend to the point of indistinguishability. (This mechanical fact also has halachic implications--see Shulchan Aruch and commentaries, Yoreh Deah, 109.) Indeed, the way to fuse solid particles to an integral whole is either to introduce a liquid element (as in the kneading of dough), or to heat them to the point of liquidity (as in welding).
By the same token, materiality tends to plurality and divisiveness, while the hallmark of the spiritual is unity and oneness. The material world presents us with a great diversity of creatures, elements and forces, each bent on the preservation and enhancement of its individual existence. The material being is egocentric in essence, striving to consume whatever it needs (or merely desires) for itself and resisting all attempts to be consumed. While there are instances of cooperation and symbiosis in the material world, these are always toward the aim of mutual benefit rather than altruistic unity; furthermore, even this usually represents a triumph of mind over matter, and must be enforced upon a resisting egocentric instinct (witness the clash of egos in a marriage, or the race and class-related tensions in a society).
On the other hand, spirituality, like water, is characterized by unity and cohesiveness, and, like water, is an agent of unity when introduced into the soil of the material. The soul amalgamates a diversity of cells and limbs into a life; the idea connects a myriad of disjointed facts into a cogent whole; love supplants the instinctive me with a common we. And when man shifts the focus of his life from the pursuit of material gratification to the service of his Creator, the diverse and belligerent granules of material life coalesce to a singular flow, as his every act and endeavor becomes an exercise in bringing harmony to the world and uniting it with its supernal source.
Beast, Fowl and Fish
The laws of kashrut, commanded by the Torah (primarily in Leviticus 11 and Deuteronomy 14) and interpreted and expounded upon in the Talmud (particularly in the tractate Chullin), establish which foods are permitted to the Jew, and which are forbidden.
In regard to the consumption of animals, the laws of kashrut distinguish between three categories of animal: a) land animals, b) birds, and c) fish. Each of these three groups have a different set of criteria for distinction between kosher and non-kosher animals. For a land animal to be kosher, it must chew its cud and have split hooves; in practice, this means that only ten species of land animal are permitted for consumption. With birds, the situation is reversed: the Torah lists twenty-four species of non-kosher birds and permits all others. Finally, kosher fish are distinguished by two signs--fins and scales.
Another halachic distinction between these three groups regards the requirement of shechitah, slaughtering. Once an animal is determined to be kosher, an array of laws govern how it may be slaughtered--the smallest nick in the knife, or the slightest deviation from the prescribed manner of slaughtering, renders the animal treif and unfit for consumption. However, these laws differ from category to category. The most stringent shechitah requirements pertain to the land animal: the slaughtering knife must cut through a majority of both of two vital passages, the windpipe and the gullet. At the other end of the spectrum are fish, which require no shechitah at all. Birds occupy the middle ground between land animals and fish: they do require shechitah, but the severing of (a majority of) only one of the vital passages--either the windpipe or the gullet--is sufficient.
The Talmud explains these differences as related to the primordial origins of these three categories of animals. Land animals were created from the earth (Genesis 1:24), and thus require a full-fledged shechitah; fish were created out of water (ibid., verse 20), and therefore do not require any shechitah; birds, which were created from a mixture of earth and water,  require the lesser shechitah prescribed for them.
What is the connection? Why is it that the earthier a creature is, the greater the need for shechitah?
To understand this, we must first examine how all of the above applies to the inner world of the human soul. Man is a universe in miniature, our sages have said, echoing King Solomon's adage, Also the world He placed in their hearts; if there are three categories of animal life on the macrocosmic level, the same is true of man--our interior biosphere also includes the land beast, the water creature, and the earth/water composite that rides the winds. Here, too, apply the laws of kashrut and shechitah, instructing us how to distinguish the desirable from the undesirable in our psyche, and how to make its kosher elements fit for consumption and metabolization in the daily process of life.
The Three Souls of Man
In the opening chapters of Tanya, the Bible of Chabad chassidism, Rabbi Schneur Zalman of Liadi establishes that we each possess two distinct souls: the animal soul (nefesh habahamit), and the G‑dly soul (nefesh ha'Elokit). 
The animal soul is the essence of physical life. Its focus is entirely self-oriented, its every act and desire motivated by the quest for self-fulfillment and self-enhancement; in this, the animal soul shares the nature of every physical being, whose most basic tendency is the preservation and betterment of its own existence. In contrast, the essence of the G‑dly soul is the striving to unite with its source, to be nullified within the all-pervading reality of G‑d. Were this striving to be fully realized, the G‑dly soul would cease to exist as a distinct entity; nevertheless, such is its nature and desire. This makes for the perpetual struggle of life: the struggle between matter and spirit, between self-assertion and self-transcendence. Any thought, desire, or act of man stems from either of his two souls, depending upon which has gained mastery over the other and is asserting itself in the person's mind, heart and behavior.
Chassidic teaching also speaks of a third, intermediary soul in every man--a soul less subjective than his animal soul, though not quite as transcendent as his G‑dly soul. This is the nefesh hasichlit, the intellectual soul. The intellect of man is the most transcendent element of his natural self, capable of objective thought and self-examination. This is not to say that the intellect is entirely free of the inhibitions of ego and self-interest; but it at least possesses the capacity to conceive of greater realities, and thus perceive the insignificance of the self before a higher truth.
The intellectual self is thus the bridge between the G‑dly soul, which strives toward a self-obliterating union with G‑d, and the animal self, which is blind to everything save the gratification of its egocentric instincts. Indeed, it is via the intellectual soul that the G‑dly soul can influence the animal soul: when a person gains a recognition of the divine truth and an appreciation of the purpose for which he was created, this very knowledge and understanding serves to refine his character and behavior.
These are the beast, bird, and water-creature within man. The animal soul of man is the land animal in man--a wholly material being, individualistic and self-engrossed as the soil from which it is fashioned. At the other end of the spectrum is the wholly spiritual G‑dly soul, characterized by the unity and adhesiveness of the water from which it derives. The G‑dly soul of man also resembles the water creature in that it lives wholly immersed in its source--just as a fish cannot survive outside of the water that spawned it, so, too, the G‑dly soul cannot conceive of an existence apart from its divine source. In the words of Rabbi Schneur Zalman of Liadi, the G‑dly essence of man, never desires, nor is it ever capable, of distancing itself from G‑d, so that even at the very moment a person sins, his quintessential self remains loyal to G‑d, taking no part in the deed--it has merely been suppressed and overwhelmed by his animal self.
Then there is the bird in man: a creature fashioned from soil and water, an admixture of matter and spirit; a creature that is capable of soaring to the most sublime heights, though it repeatedly returns to earth to rest and feed between flights. This is the intellect of man, capable, on the one hand, of raising itself above the materiality of earth and attaining a higher vantage point on life and self, yet nevertheless bound, in many ways, to the physical reality of which it is a part.
Before an animal can be eaten, to become the stuff of our bodies and the motor of our lives, two conditions must be met: it must be determined to be kosher, and it must undergo shechitah as dictated by Torah law.
Shechitah is only to draw forth, states the Talmud (Chullin 30b). The most basic meaning of this rule is that the slaughtering knife must be drawn across the vital passages--pressing downward, or other deviations from the required back-and-forth movement, disqualify the shechitah. Chassidic teaching, however, uncovers the deeper significance of this law: that the function of shechitah is to draw forth--to draw the animal out from its beastly state and into the domain of a life consecrated to the service of the Creator.
This is achieved by slaughtering the beast--i.e. taking its life. The material world is not, in itself, a negative thing; what is negative is material life--the passion and zeal for things material. The Jew knows that the reason that man has been granted mastery over the physical world is to utilize it in his fulfillment of the divine will. Man was created to live a spiritual life that is sustained by the material, not a material life which his spiritual prowess has been harnessed to serve; to crave the physical for its own sake is to become part of it rather than making it part of you and a partner to your transcendent goals. So even after man has separated the kosher aspects of life from the non-kosher ones, rejecting all that is irredeemable and corrupting, he must still slaughter the material beast before it can be consumed. Only after its life has been taken out of it can it be sublimated as an accessory to the life of the spirit.
Hence the differing shechitah requirements for the three components of the inner life of man. The animal soul requires a full-fledged shechitah: comprised solely of the soil of materialism, it must be drained of all vitality and passion so that its substance might be drawn forth into the realm of holiness. The intellectual soul, comprised of both soil and water, requires a partial shechitah--its material and egotistic elements must be subdued, but there remains much about the intellect that is desirable in its animated form. (This is also why there are more non-kosher land animals than kosher ones, while the reverse is true of birds.)
Finally, the wholly selfless, wholly transcendent G‑dly soul requires no shechitah at all, for both its substance and spirit are desirable and digestible elements in the life of man.
1) Fish... do not require shechitah; rather, the act of gathering them (i.e. drawing them out of the water) is what renders them permissible for consumption. For the Torah (Numbers 11:22) says: 'Might sheep and cattle be slaughtered (yishachet) for them... or might all fish of the sea be gathered for them'--implying that the 'gathering' of fish is the equivalent of shechitah of cattle and sheep (Mishneh Torah, Laws of Shechitah, 1:3).
2) In 1:20, the Torah counts birds among the creatures spawned by water, while 2:19 lists them among the creatures G‑d created from the earth. Hence, the Talmud concludes, birds were created from a mixture of soil and water.
3) The concept of a good inclination (yetzer tov) and an evil inclination (yetzer hara) in the heart of man abounds in the Talmud and the Midrashim (cf. Talmud, Berachot 61a). What is unique about the Tanya's thesis (which is based on the teachings of Rabbi Chaim Vital, a disciple of master kabbalist Rabbi Isaac Luria) is that it speaks of two souls--two entire personas, each with a full set of traits and faculties. The two inclinations are actually the drives and desires of their respective souls.
4) Nevertheless, it does require gathering, as per note #1 above. For the G‑dly soul's decent to earth to animate a physical body means that its prowess are fragmentized by the diversity and divisiveness of material life, necessitating an internal consolidation and a rallying of forces on its part.