In which I will post information as I find it!

CHI
There are a ton of different philosophies, but I'm rollin' with the ones that seem most relevant.

Chi is essentially a life force that permeates everything.  It cycles through the human body, it connects humans with others and their environment, and, if nurtured, one can use one's chi to affect the universe.
- Wiki gives a nice overview.

In the Avatar 'verse, chi is used for more than bending.  It can help healers who direct chi through certain channels in the body; the spiritually-inclined can 'tap' into chi to locate other living organisms.  While it's not mentioned if that skill can be used on nonliving things, it is established that chi connects everything in the universe.

Living things (even clever animals) can have a chi-based connection with another living thing.  This connection allows the two beings to locate each other and even get a read on their thoughts and emotions by tapping into their chi.  So only benders might be able to manipulate the elements, but even nonbenders can have some control over chi.

Chi blockers can interfere with chi by hitting weak points in the channels between chakras.  The technical details are glossed over.


TAI CHI (or, more correctly, t'ai chi)
There are different forms of t'ai chi based on different aims.  Most forms are slow, but some popular styles rely on speed.  At its base is the philosophy produce great effect with minimal effort; this is thought to be done by manipulating chi (making t'ai chi stylistically similar to bending in a number of ways).

T'ai chi exists in many forms and those forms differ significantly.  Generally, though, they all stress the importance of balance (yin and yang, sport and health, etc), memory, coordination, breathing, and some degree of meditation.  It's an essentially peaceful art that advocates against fighting brute force with brute force.  Instead, practitioners are to go along with the offensive movements of their opponents and redirect the energy their opponent expends into an attack of their own. 

T'ai chi forms focus on establishing and maintaining a center of balance and going through the full range of natural motion.  Posture and form are important as well.  The style is open-handed and not meant for combat; instead, its purpose is to promote relaxation, strength, flexibility, and technical mastery of practical moves.  Learning is slow, forms are long, and repetition is inevitable.  A significant part of t'ai chi focuses on the breath.  Chi is air, remember!  When moving, one breathes in as one collects energy (both from internal sources and the air) and then out as chi is directed into a move.  Sweeping and circular motions are very common, as both are thought to build chi.

(It's easier to imagine if you think of chi as something that's sitting around in the air.  Breathing it in gives us power; we release power when we breathe out and strike.  The timing of breath is important because you can't direct your chi if you released it in the build-up!  The sweeping and circular motions are meant to gather chi.  Sometimes it's helpful to think of chi as a ball that grows as more air-energy is added to it.)

The repetitious nature of t'ai chi is what gives it its meditative and calming quality.  Yes, you're calling upon a lot of muscular strength, flexibility, balance, and awareness of breath when practicing, but the goal is to repeat until it's second nature.

T'ai chi sparring is all about redirecting energy and balance.  Defensive or neutralizing skills are the most important and teaching starts out slow.  Practitioners try to find their opponents' center of balance and capture it, using that offensive energy against the opponent.  Slow, meditative sparring allows practitioners to learn how to best use an opponents' strength to their advantage, maintaining balance while throwing others off of theirs, and anticipating attacks.  Mastering all of this makes a student better at applied martial arts.

I don't recommend poking around the internet too much.  There's so much information out there, so many styles, and... it's overwhelming and hard to grab useful facts.
 

CHI AND BENDING IN THE AVATAR 'VERSE
All four classical forms of bending are dependent on chi, specifically the chi in the human body.  Bending is basically the manipulation of one's own chi to affect and interact with the environment.  It's all about redirected energy.  Does that sound familiar?  Like, oh, t'ai chi?

Due to their inherent spirituality, all Air Nomads are born airbenders.  There's no rhyme or reason to the distribution of bending abilities with the other elements.  Each element also has its share of strengths and weaknesses, maintaining a balance in which no element can overpower the others.


AIRBENDING
Airbenders are the most spiritually attuned and the most detached from the physical world.  It's an art that developed peacefully and it relies heavily on flexibility and following the path of least resistance.  Many airbending moves are defensive in nature rather than offensive.  Go here for specific airbending moves.  Weapons like staffs can be used to channel and direct chi more accurately.

Airbending is passive and employs evasion and mobility; a lot of the art is movement, dodging, and simply avoiding engaging with the enemy.  Spirals and circles are prominent in the culture because they represent the way an airbender moves, changing direction as required.  Airbenders' attitudes largely reflect their bending methods.  Additionally, airbenders tend to be very agile and light on their feet.  Intuition and imagination are prized because these skills allow for more freedom... which is what airbending is all about.

Significant weaknesses with this art include its practitioners' aversion to violence (yes, this is a weakness) and ineffectiveness in an enclosed area.  An airbender who can't move can't dodge or use the sweeping movements often preferred.


WATERBENDING
Waterbending has numerous styles, making it very flexible (fitting, since water is considered an element of change).  It's considered the most adaptive of the bending arts.  Waterbenders interact with their environment to a large degree and their art is intimately connected to the moon and ocean--to a point where damage to the Moon and Ocean spirits is detrimental to waterbenders.  This art is generally fluid and graceful, not to mention versatile.

Waterbending employs defensive moves that can also be used on the offensive.  Waterbending focuses on control and using an opponent's own attacks against them.  Waterbenders are at their strongest at night and with the full moon.  Their movements are typically graceful, circular (not to the degree of airbenders), and slow.  Methods of defense and attack are many and varied, so go here to read about them.

A weakness of waterbending is its relative slowness and the fact that it's not a strike-first kind of art.  A waterbender's attachment to the environment also gives them the ability to inadvertently make freaky things happen when their emotions get out of control.  Waterbenders are also limited if they can't use their arms, as that's where most of their motion is centered.  They're vulnerable during lunar eclipses.

Southern-style waterbending tends to be more aggressive than other styles.


EARTHBENDING
Earthbending emphasizes strength, endurance, and neutral jing (power).  It involves listening and waiting for an opportune time to strike--and then striking forcefully and unrelentingly.  Earthbending tends to be diverse because individuals are likely to develop their own styles.  Pro-bending utilizes a different, faster kind of earthbending that involves less waiting and a lesser degree of rootedness.  Earthbenders also use their neutral jing to anticipate their opponents' next moves.

Earthbending is equally offensive and defensive--balanced, rooted.  Its practitioners must be decisive and unyielding.  This art, like water, relies on a strong connection to the element via chi.  Earthbenders can amp up their power by fighting barefoot.  The vast majority of earthbending moves rely on the lower body.  Here are some techniques!

Weaknesses include wood and metal.  Wood is non-mineral and not all earthbenders are skilled enough to bed metal; some metals are too pure for even the most experienced bender.  Incapacitating the lower body also negatively affects earthbenders, who are at their best when they're firmly connected to the ground.


FIREBENDING
Firebending is extremely aggressive and the element is generated from within the user's body, not pulled from the environment.  Fire is the element of power and bending relies on force and tenacity.  Firebending, because of its aggressive nature, doesn't involve a lot of defensive maneuvers.  Stylistically, it varies largely according to circumstance (for example, methods will be different in a small space than in a large arena).  Firebenders draw strength from the sun and even comets; they're strongest during the day.

Firebenders use their own internal heat to generate fire.  Their movements are typically fast and furious.  Circular movements help a firebender build power, but their circles tend to be small and tense in relation to, say, those of waterbenders.  Firebenders draw energy from heat sources and are at their best in warm climates.  Skilled firebenders require control; fire is an unpredictable element and must be channeled properly.  Check out moves here.

Cool temperatures decrease the power of a firebender, as can rain.  An inherent lack of defensive moves can be a weakness.  Firebending requires a spiritual "inner fire" and, in practice, drains vast amounts of energy, making firebenders sprinters rather than marathon runners.  To master firebending, a bender must have considerable breath control and self-discipline. 

Firebenders are less powerful at night and during solar eclipses.


CHAKRAS
Chi pools in certain points of the body; these are called chakras.  There are seven chakras and each is associated with different moods and functions.  Chakras can be closed or open; when open, energy is allowed to flow freely through the body.  Emotional debris in the channels connecting the chakras can, in the Avatar 'verse, negatively affect bending or block it entirely.  Opening all of one's chakras is a lot like attaining Nirvana: connections to the earthly world need to be severed to have full access to the spiritual.

(Avatar 'verse is a little confusing in terms of this.  Aang can only reach the Avatar state by unlocking all seven chakras, but other Avatars have entered the state while still nurturing attachments to the physical world.  It's also stated that the Avatar is always a mortal human because it's necessary to experience life and emotion to know how precious both are, giving them more reason to protect both.  It should, therefore, be impossible for the Avatar to truly detach from the physical world, but whatever.)

There are seven chakras.  Their importance in Avatar Land is covered rather thoroughly here.


MENTAL ILLNESS
Post-traumatic Stress Disorder

"PTSD is a potentially debilitating anxiety disorder triggered by exposure to a traumatic experience such as an interpersonal event like physical or sexual assault, exposure to disaster or accidents, combat or witnessing a traumatic event. There are three main clusters of symptoms: firstly, those related to re‐experiencing the event; secondly, those related to avoidance and arousal; and thirdly, the distress and impairment caused by the first two symptom clusters." (PubMed Health)

- Public Medical Health PTSD overview
- National Center for PTSD
The second is an excellent source and even goes into specific kinds of trauma.
- Mayo Clinic
Sort of watered-down, but it has the most concise definitions, I think.  Good links, too.

Depression and anxiety are huge parts of PTSD.

If this isn't thorough enough, lemme know!