Ahh I love this type of discussion, please excuse my enthusiasm.
I teach relaxation almost in every single lesson with my students of all ages and of all levels. Often the relaxation comes from considering fingering and subtle differences in doing it right and wrong. From mastering our fingering in passages one can then develop more relaxation through the entire body. Liszt even says "Technique is fingering and returns to it."
Incorrect fingering causes all sorts of trouble through the entire body. And the problem is that although you might hit the right notes with the right fingers your fingering may have subtle inefficiencies. The way to correct your problem is to focus on the fingers and hand, not so much the other parts of the body. If the other parts of the body are merely loose and not tight that is enough, focus on your hands and fingers.
When I play the piano all I am thinking about is the hands and fingers as well when it comes to what I have to do. I hardly think about my arms, shoulders, back, even the pedaling that breathes with the piano you hardly think about the foot! But the hands and fingers, generally your body's connection to the keyboard, this is what you constantly feel changing forms and gives you the physical feeling of pleasure playing the music! I feel the rest of the body reacts with the hands and fingers not the other way around when playing piano.
As relaxed as possible.
Using the the right muscles at the right time.
Generally relax the arm and shoulder and relax as much as possible. It's ok to firm them up a bit when they're needed, sometimes like a pulse of firmness to play the notes.
This is important point Bob has brought up "controlled tension". To say everything you do is always relaxed and without any effort is a lie because otherwise we would be able to play with full force at the age of 90, which is not the case. When play loud forte ranges we must be able to direct energy through parts of our whole body to create the sound. If we have rapid position changes we need the ability to constantly turn on and off the tension, which taxes you no matter how good your technique is, but the better you are the more you can do it with less effort but produce the same sound, which in the end increases your chances of producing the desired sound and hitting all the right notes.
How on earth can we describe it in words what controlled tensions is like, I can only imagine saying: Flicking with a push immediately followed by a release (which can keep the fingers resting gently on the notes or withdraw them). The energy can come from all parts of our body but is felt draining through the hand/fingers. I generally visualize: fingertips as (pppp), 1st knuckle (ppp), 2nd knuckle (pp), 3rd knuckle (p), under palm (mp), above palm (mf), wrist (f), forearm (ff), shoulders (fff), back (ffff). When we produce the volume dynamic or accent we feel the energy coming from these parts of our body but it is draining through our hands/fingers, so what is happening here is most important and will control the rest of the body, not the other way around.
I always tell my students to feel lazy when they play, when the body does not need to be activated or tense do not let it be. Sometimes this is very difficult for a student to notice while they are playing that is why a teacher is helpful to direct your attention to subtle points that you might otherwise play over.
How a student sits is often an important issue commanding relaxation. I demonstrate this to all my beginners by getting them to try and play their left hand while it crosses the center of their body then asking them to give their LH more space by moving their center more to the right (or by leaning back slightly if they are forced into crossing their center with the LH), then things become more free. There are many more issues much more subtle than this one though, but if certain issues are not taken care of or noticed it may be quite difficult to acquire relaxation while putting yourself in disadvantageous positions.
Most relaxation problems however I find are from the fingering of the notes. As a teacher I focus on getting the students to master their fingering, this then almost always controls the rest of their body. We indirectly change how the rest of their body moved by efficiently solving how our hands/fingers should move at the keyboard.
I think it's more about being AWARE of what your body is doing. There's no such thing as actually being completely relaxed when you play piano, it's impossible, but what you want is to have the minimum amount of tension needed to execute a particular passage.
I think points of relaxation recover more energy than points of exertion take from us when we play a piece. If we cannot produce this feeling while playing a piece the piece tends to tax a lot of our energy and things can become more uncontrolled. Of course there is nothing wrong with a little bit of lactic acid building up when you play long strenuous passages, but how do you release that tension and recover from it? Fortunately there are not many long pieces where it is 100% maximum output from start to finish, there are always points of recovery.
It is not good just thinking that just because a passage is difficult and is supposed to tax you that after we escape it we must carry on the feeling of tiredness and exertion. After points of exertion we know how to relax and recover our energy fast so that our muscles do not get tired. Growing up on a very heavy action grand piano I had to learn how to conserve energy while playing from the beginning. This type of skill has passed on now that I learn on lighter action pianos but play pieces that demand much more technical acrobatics which our body must try to solve efficiently first with good technique and then secondly learn to relax after this exertion if we are to be able to play at that standard with effortless mastery.
If you did go completely relaxed at the piano, you could still pull off playing a cluster of notes FFF with your forehead one time when you tip over.
lol, use of gravity and natural body movement is a friend for us pianists, this is a good point!
I like rachfan's responses,
....especially relaxed arm weight.
...The idea is NOT for the arms to be so totally relaxed that they go into free-fall due to gravity, thereby allowing the hands and fingers to crash into the keyboard. Relaxed arm weight actually needs to be directed by the upper arm (for accuracy), and secondly there has to be just sufficient enough control during the descent that the hands and fingers drop and sink into the keys so as to produce a rich tone.
This is very important for playing patterns that move positions especially chords but not neglect other patterns of movement. What is important is that the connection we make with the keyboard which is made with our hands/fingers activate a moment of muscular memory form to control the flow of this energy, then we deactivate this movement in our hands, relax, and allow the relaxed arm weight to take over. But the arm weight even is guided by the feeling it produces into our hands/fingers, so again this points many times to our hands/fingers being the most important points of consideration.
Key velocity which transfers to hammer velocity is the main element in tone production. But it goes deeper than that. Consider a would-be pianist who brutally forces the keys down leading to fast hammer velocity yet results in unpleasant "pounding" or "banging" as we say. That is not a rich tone! If that same person is then shown how to relax the arm and allow it to descend with gravity--but with a modicum of control for accuracy in depressing--not crashing into-- the correct notes--the banging will, with practice, be replaced by a rounder and richer tone. So there is a bit of deliberate tension mixed in with the relaxation, which is why "relaxed arm weight" is figurative, not literal. Obviously this is not an objective scientific principle; rather, its more subjective. It's not so much quantitative as it is qualitative. But I believe it to be essential in attaining a beautiful tone.
Someone who reads an intense forte marking in a score and merely pounds at the keyboard with no consideration to what they have previously played, are playing and will play, merely plays with technique out of musical context. If we understand the music we are playing then we will know how to bring our instrument up to certain tiers of volume. To merely pound at the piano highlights little understanding of your volume control, unless the note you are playing needs to be the loudest note you can possibly produce there should be no need to pound the piano. So pounding has its place but only if you need literally the maximum volume possible, this is not called for in most pieces written and merely highlights uncontrolled playing much more often than not.
Sensitive players certainly feel and understand all the shades of volume pppp through to ffff and beyond. Not many pieces are written which demand such a broad spectrum of volume shading but certainly there are some which focus on shades of p or f in sections which really highlight if someone knows what they are doing or not. Rachmaninov can have such shades of forte but the lesser pianist will come off as pounding or harsh, the elegant one will pull off the different shades and seem to get the volume tiers just right making the climaxes simply explosive (even more so than pounding which might produce more dB!).
Back to key/hammer velocity for a moment, Ortmann believed that the same key velocity achieved by a finger, pencil, or umbrella tip would create the same tone. He was probably right so far as that statement goes......But the major fallacy is that the pianist plays not a single, isolated tone all the time, but usually many tones including chords.
He has a point but it certainly limits the amount of music you can play

I think it highlights a point that we should not get overly interested in what a single finger does but what a group of fingers produce in the whole phrase of music. Some people can get overly interested in how to bring out a single note melody while playing so much so that they do all sorts of strange things with their fingers to make that note stand out, we should always see things connected in context with each other, not separate them and look at them individually. I think that might be what Ortmann was on about also. I have also caught myself doing strange things to bring out a melodic note sometimes and have to correct myself from doing these unnecessary mannerism to draw the note out (I can only describe it in words like keeping a certain part of my fingers tense which are producing melodic notes while they hold the notes or move to another while all the others are relaxed, although I have played pieces which require this type of tension and can produce it without problems it often is unnecessary and merely isolating my thinking without looking at the whole picture).
.... During a grueling practice session, who here has never experienced a buildup in tension affecting the whole playing apparatus? Probably nobody! If ignored, it will soon spread to the neck, which will make the discomfort even more noticeable. While tension builds, concentration, accuracy and artistry are all diminished. Now if the pianist gets up, moves away from the piano, swings the loose arms parallel to the body, then swings the arms in front of the body such that they cross one another forming an X, and then bends over from the waist a bit, dangles the arms while shaking and rotating them like two ropes being blown randomly about by the wind, like magic all tension is released immediately, and back at the piano relaxed arm weight is resumed leading to restored artistry. So this is an empirical effect that we can directly observe and feel thereby informing us that relaxation benefits artistry.
This is another way of describing how points of rest replenish more energy than points exertion will ever take from us. Our body will seize up and refuse to go on if we exert ourselves too much, then we recover energy quite fast and can carry on. Most beginners who learn the piano will have to contend with feeling tired and worn out. You challenge yourself to play in a way which prevents it, learn to recover from it and try again. But you need to actually have this worn out feeling to know what it is like so you can work against it. Even when I study new pieces when I play certain passages I will feel tension creeping in and it prompts me to investigate what the culprit is. So tension although it should be avoided, acts as a compass steering us in the right direction.
One qualifier: Although the fingers are part of the playing apparatus, they can NEVER be relaxed. Instead they always need to be taut, otherwise articulation will sound more like wet noodles or cotton.
There is usually an exception to every rule, and the fingers are the exception to relaxed arm weight. Is it a paradox? Yes, but life is a paradox too.
If we totally relax our hands our fingers will curl up in a fashion that is not really suitable for piano playing. So merely extending our fingers to place them on the keyboard is more tense than being totally relaxed. Then to be able to sit up right, to have our feet on the pedal, butt on the edge of the seat etc, these points cause more tension to our body than our fingers merely resting on the keys. So in relation to the rest of our body while we sit at the keyboard, our hands resting on the keyboard is without tension. Your back will get tired much faster before your hands get tired resting at the piano. Just sit at the piano with your hands resting on the keys for 1 hour without moving and see what part of your body get tired first

My back go first, soon after my ass starts getting numb before any of my main piano playing apparatus ever start failing!