In waltz, some teachers said body weight should be on beat ( 2nd beat) totally., some teachers said body weight should be between 2 feet ( make "A" shape ) shift weight when count 3, I really confused.

"Some teachers" can say about anything, really. It doesn't particularly mean they are right or wrong; they just have a different method of delivering the message. But, I would be confused if it was explained the way you summarize in your note. I was always taught that the hard emphasis or total body weight, if you will, is on the first step, because the weight lessens on the rise of 2 and 3. I think the second part of your note makes sense, as you definitely have to shift weight when completing count 3. Even on this site's learning center, it is made pretty easy to understand when it states "Strong accent on 1."
There may be some other approaches to this, and the long-time instructors on this site always provide valuable insight as to exactly what your instructor may be trying to say, or if there is something to that interpretation. Regardless of how you want to think of weight pressure and shift, I have always just thought about rise and fall. If you have that concept stuck in your mind, and your body is following it, I would think the weight shifts would occur the way they should.

Imagine a ruler that spans from 1 to 3 with equal amounts of distance between the numbers. Imagine the distances in terms of time between notes. We begin our first step on 1, and we have all of the time encapsulated in that beat to complete the step and transfer the majority of our weight over / onto the moving foot. We transfer some of our weight initially, then more weight, and finally most of it before the next step begins. The trailing foot will begin to skim over the floor to meet or pass the first foot (depending upon what figure is being executed) at the begining of count 2. As the second foot skims over the floor, it is providing balance by supporting some of your body weight. By the end of count 2, "most" of your body's weight should be on the new, supporting foot. The same process begins again at the beginning of count 3 and so on through your dance. And, depending upon the musicality of the song and the expression of the dance, there may be variations of the movement and weight change. Your body weight moves in time and across the distance between the begining of the notes. Try not to focus too much on the end of a note, as this might impede a smooth flight of the body across the floor and through time because no sooner do we arrive must we be ready to continue in our movement using the weight over the supporting foot to provide traction allowing the muscles to create movement. Rarely should all of your weight be over just one foot in ballroom and rarely, and particularly in the smooth dances, should we not have both feet in contact with the floor beneath us at most times. I find all of this to be a relatively easy concept to understand, but as of yet, I dont have it fully mastered.

Unfortunately this is a topic likely to become useless complicated.

Probably what is of most initial importance is to realize that a typical waltz figure having three steps has three complete weight changes, and must be executed smoothly.

Beyond what is of practical initial use, some objections to what has been posted:

"We begin our first step on 1, and we have all of the time encapsulated in that beat to complete the step and transfer the majority our weight over / onto the foot"

To clarify, for waltz, many people would think it more musical to land either the foot or the body weight (which just about immediately follows) on the beat, which means the action of the step will have to start beforehand, during the previous beat.

"As the second foot skims over the floor, it is providing balance by supporting some of your body weight."

This is generally not advisable, as it means you will have a lot of trouble dancing on a floor that is stickier (or for that matter more slippery) than the one you are used to. It's generally better to be precise with the relationship of the body weight in respect to the standing leg alone. But there are a few places where a little pressure from a foot that you have departed _but not yet moved_ can be useful, at least when initially learning.

"Rarely should all of your weight be over just one foot in ballroom"

Actually, most all the time your weight should be supported by only one foot. That doesn't mean your body is necessarily located over that foot, only that it's the only one holding you up. This can readily be seen by stepping through the frames of just about any video of top dancers. At least in the waltz and foxtrot, you won't find very many stills of moving figures where a credible argument could be made that the weight is divided.

However, despite not being the way we ultimately dance, it can be a good strength training exercise to slowly shift the body weight from one foot to the other over several counts, particularly to execute a controlled lowering with the feet apart. And sometimes the versions of dances initially taught are more exercise drills to prepare and condition the body than artistic movement.

"and particularly in the smooth dances, should we not have both feet in contact with the floor beneath us at most times"

Freely sliding in contact or near contact, yes.

I don't subscribe to the 'divided weight' theory at all.

We change weight fully on each step. On a forward walking step (count 1 in Waltz) it is OK either to have the moving foot very slightly off the floor, or to just skim it, but this doesn't amount to 'pressure', and it is the use of a heel lead and body shaping that gives this step the accented emphasis. On the side and closing steps, it is usual for the moving foot to skim the floor, but very lightly.

It is worth looking at one of the standard texts closely regarding the mechanics of the ballroom walk action. There is a very brief moment when as the forward foot has been placed on the floor, body weight is equally divided between the heel of the front foot and the ball of the rear foot, but as the body is in movement, weight transfer is almost instant, and the next step cycle begins as soon as the rear foot, once it is in movement, passes the standing foot. The same thing occurs on backward steps. In Side or closing steps the mechanics of the movement prevent such an obvious 'moment' of weight change, but it is present, even if it is slightly prolonged.

The only dance where weight is routinely divided between the feet is Argentine Tango (and then only ever for the man). In Ballroom, we aim to have substantially our whole weight over one foot or the other at all times.

I'd like to make a few comments in the hope of explaining a couple of things that might cause confusion for the original poster Judi.
There are beats to the music, and there are starts to 'steps' as defined in the technique book. I believe that the technique books do not precisely define whether or not the starts of the steps coincide with the beats (if there is such a definition, please somebody let me know). However, most people i see (including myself) would dance a 1,2,3 of the natural turn so that the first step (right foot forward for man) plants on beat 1.
Once the right foot has planted (i.e. the heel has stopped moving, or toe of left foot for lady) it becomes the only supporting foot, apart from a brief moment (as Telemark has said) where the weight is transferring from the previous standing foot (in this case, the left) to the new standing foot. However, i think it might be a bit confusing to say that our "whole weight is over" the new standing foot. This could imply that the body is over the standing foot. The body moves over the standing foot (at roughly beat 1& in the music)but it's moving smoothly and moves 'in front' of this right foot as we progress to step 2.
The moving foot only skims the floor, but can provide a small amount of balance if you need it. This is particularly useful between steps 2 and 3 when the feet are closing (in this example). But don't think of it as supporting any significant weight.
Sorry to be so long-winded. It would be so much easier to show/demonstrate on the dance floor.

The start and finish of a step, as defined by authoritative sources such as Howard is when one foot passes the other. Unfortunately, this gives rise to considerable practical problems in interpretation of some charted figures and the actions that they describe, and that is not a topic for the feint hearted.

However, there is not really any contradication in saying that one foot is supporting the body (except for the split-second of transfer from one foot to the other), and recognising that in movement the standing foot is only 'under' the body, supporting the weight directly over the foot for a small proportion of the whole step. If we had the physical ability to stop dead, at most points in any step, we would immediately fall over. We do use the moving foot for balance and control, but rarely to support any significant weight.

Jim Maranto (twice American Smooth champion) says your weight is usually divided between both feet, in varying proportions; seldom is all your weight entirely on one foot. For one simple example, consider a forward step in Waltz or Foxtrot. "Push off on the supporting foot." The foot you stepped on last supplies the power to move the other foot forward, and you transfer weight onto the moving foot. You do not have all your weight on one foot one instant, and all of it on the other foot the next instant. Weight changes, yes. Instantaneous weight changes, no.

".... says your weight is usually divided between both feet, in varying proportions; seldom is all your weight entirely on one foot."

Then, like most who haven't devoted a fair portion of their career to its perfection, he's not very expert at essential ballroom movement.

There are also a number of people out there who say things in conflict with what they actually do. Sometimes they are recommending a stepping stone exercise rather than the ultimate goal, sometimes they really don't know about the inconsistency - and I suppose sometimes they fall short of their own recommendations.

"For one simple example, consider a forward step in Waltz or Foxtrot. "Push off on the supporting foot." The foot you stepped on last supplies the power to move the other foot forward, and you transfer weight onto the moving foot. You do not have all your weight on one foot one instant, and all of it on the other foot the next instant. Weight changes, yes. Instantaneous weight changes, no."

Actually, if you watch video of highly skilled standard dancers where movement quality is stressed much more, it pretty much is instantaneous. Driving steps get drawn out to the point that by the time the moving foot actually stops, the standing foot is pretty much used up. In smaller steps, especially sideways on the inside of turns, you can see the departed leg immediately relax - even though it could continue to do something, that's not the desired character of movement.

An exception would be tango, where at least some interpretations keep more pressure in the departed foot until it actually moves, due to most steps lacking flight through the arriving foot. And of course there are some split-weight figures, like the contra check.

Judi. As a lady your weight is different to the man. The lady on step two, (whilst still on step one NFR), Step two at that point becomes a point to the side. The weight only arrives after the man who does have foot rise on step one goes past and you follow him up