What is a Theory of Everything (TOE)? I see many learned contribution about the role of a Theory of Everything (TOE), what it
might do or not do, what kind of answer it might provide, and what
not.
But I do not know what a TOE is, how I would recognize it if I met it
in the street. Is it an axiomatic theory, a collection of equations, a
sacred book, an incoherent dream, a good topic for fun discussions on
a forum, or just a unification of QM and GR, which might not even tell
us everything about either.
It would be nice for naive users like me to know what is being talked
about, and whether it is the same for all who talk.
Personally, I naively took it to mean a complete description of the
physical universe, which does remain very vague, and may not be
meaningful.
 A: 
Personally, I naively took it to mean a complete description of the physical universe, which does remain very vague, and may not be meaningful.

I can describe for you the mathematical form of a theory of everything:
       #=0

Where # is a symbol which includes convoluted differential forms. This is the form all partial mathematical models of electromagnetism, mechanics, thermodynamics ... every type of mathematical model for physical systems has taken in the past and will appear in the future. They can be reduced to differential forms on the left and a zero on the right.
It will have its mathematical axioms that will make the solutions rigorous but it will have the physics postulates that have to be satisfied as the  input in choosing a particular TOE from multitudes of similar ones.
The word "theory" in physics does not simply have the burden of mathematical rigorous existence. It could be very rigorous mathematically and irrelevant for the physics to be modeled. Mathematical theories become models for physical states.
The fact that it will be mathematically rigorous means that the solutions will describe correctly all known physical data and predict, given the boundary conditions, any new ones we could think about, in precise numbers. There is nothing vague about it.
We expect that the Standard Model, which describes almost completely all known up to now particle data, will naturally nest in the model of the TOE. There is nothing vague about this either. We expect gravity to be modeled naturally within TOE.
At the moment it seems that string theories offer all these options, but as there are thousands of possibilities, the final model has not been found yet, not even the class of models within string theories, which can be candidate for the embeddings necessary of the SM to assure consistency with existing data. If/when decided upon the predictions of the model will be tested for consistency with new data. 
A: I think a good way to start is to understand what a theory is. A theory is a mathematical model. This sounds complicated, but it's generally just a set of equations that we solve to find out how our system behaves. So Newton's laws are a mathematical model that describe the motions of particles at velocites low compared to the speed of light.
So far at least, all our mathematical models are approximations that give accurate results only in a restricted range of circumstances. So Newton's laws apply only at speeds low compared to the speed of light and fail when the speeds approach $c$. Special relativity gives accurate results for all speeds, but fails when gravitational fields become high. General relativity fails (probably) when the density is so high that it predicts singularities.
The reason all our existing theories fail at some point is that we have made approximations to simplify the theory, and there will inevitably be circumstances under which our approximations aren't justified. It would be nice if we had a theory that didn't rely upon any approximations. Such a theory would never fail, so it would in principle describe everything in our universe and possible others. This would be a theory of everything. Note that you wouldn't actually attempt to use the theory to describe anything any more than we'd attempt to use QCD to understand hydrodynamics.
Initially it was thought that String Theory might be theory of everything, but then we discovered that it was (probably) a low energy approximation to M Theory, so it ends up being an effective theory just like all the others we have. I don't think enough enough is known about M Theory to know whether it is a possible TOE.
The appeal of a TOE is that because it doesn't require any approximations it would be a true representation of the universe(s) i.e. we'd understand how the universe really works (whatever real means in this context). Every now and then you hear a despairing cry that such a theory will never be possible and all we can achieve is increasingly good approximations to reality. I suspect most of us think this is unnecessarily pessimistic.
A: The ToE in a way is the surpassing of
1) Current knowledge.
2) Current errors.
3) Current isolation.
Things that are yet to be learnt can put a halt on progress since change is required to move from the old to the new. Current errors can be tough to overcome if these errors have been overlooked for a significant time period. A common component can be viewed from numerous directions and thus be seen in numerous ways, thus be seen as separate components even though they are not separate. Again, this too can be tough to overcome if this separation or isolation has been overlooked for a significant time period.
Also, to see and understand the big picture concerning reality, yet be doing so while still being inside the very same reality of which you try to understand, another barrier must somehow be overcome.
In short, you may find the equation that takes the EVERYTHING into account, however the human mind may not still understand the equation due to the act of understanding meaning that the mind must wrap around that which it is to completely understand. 
Thus the mind must wrap around the reality of which it resides within to thus completely understand it.
Thus we are confined to merely the title of the "THEORY" of everything instead.
