Every day we find ourselves having to face the various aspects related to energy saving, we often use terminology that lends itself to non-univocal interpretations. We do not always dwell on the real meaning of a term, many times, we draw on "hearsay", but not always  we check the sources or, better, we do not always have the time to dwell on  think about what we are told.

Breathability
From Wikipedia:  In construction and in materials science, the term breathability of a material (in English "breathability") refers to the ability of a material (for example a building material or a textile material) to be traversed by humid air. Breathability is generally related to the porosity of the material.

The advantages of perspiration
The more breathable a material, the lower the  possibility of condensation on its surface of the material.
The breathable power therefore determines greater durability  of the product, as the water that would be formed near the surface of the material, would make it subject to greater temperature changes and more easily perishable.
Breathability also allows for better thermal insulation,  in fact the air in stagnant conditions (i.e. in the absence of convection) is a good thermal and acoustic insulator, but it would lose these properties in the presence of  liquid water, which instead has a high heat exchange coefficient.  Furthermore, a high transpiring power of the material favors the recirculation of oxygen between the external environment and the object under examination (for example the  living place or a garment).
Perspiration also plays an important role in the choice of  clothing fabrics: more breathable fabrics in fact ensure that the humidity of the human body is removed more easily, thus decreasing the condensation of sweat, and increasing comfort. The transpiration of the tissues therefore favors biological transpiration. [1]  In the case of fabrics used in clothing,  materials are often required to be waterproof and breathable at the same time.

Note
Although human transpiration is favored by the transpiration of  tissues, the mechanisms of these two phenomena are quite different: in fact human transpiration has a biological nature and has the purpose of lowering the body temperature, while the transpiration of materials is a phenomenon  chemical-physical which is simply linked to the passage of air and humidity, and does not substantially alter the temperature of the material.

Considerations
This definition is perfect, incredible and perfect. In a few  words touched on all points of the question.  If they hadn't entered the "Note", we should have  we. Here is all the misunderstanding that underlies the claims  science fiction about the breathing of the house made through the walls.

We explain  better:
it is absolutely true that excessive humidity inside the  dwellings is determined by a high production of steam by the inhabitants (cooking, washing, putting the laundry to dry on the radiators, etc.). The possible existence of rising damp must also be entered here  capillary that releases water, in the form of steam, inside the  locals.
An average but indicative value is a production of 10 liters of water (in the form of steam) per family. To restore the state of well-being and to avoid damage such as condensation and therefore mold, bad smells but also deterioration of the structures, the excess water or, better, the humid air must be disposed of. Less than one percent manages to be disposed of by diffusion (this phenomenon is similar to what happens in the pulmonary alveoli, the analogies stop here, in the use of the same term), let's also call it transpiration through walls and roof.
There is no doubt that 99% must be eliminated by exchanging the air present in the premises. How?

In the simplest way: opening the windows, Mrs. Maria would say, and having a suitable extractor hood in the kitchen!

Obviously that's not the only way heat exchangers are  excellent tools, already known in antiquity, to maintain a perfect  hygrometric balance, without significantly wasting  power.
There are many systems for having mechanical ventilation  controlled, the important thing is to become aware of it, trusting Mrs. Maria  or technology, both systems are fine, as long as we leave  lose the commercial fantasies that have made us believe, for years, that we have  need for the "breathing of the walls"!
It is evident that throughout the day, the presence of relative humidity varies with the variation of human activity. In the Hygiene courses  Environmental, it is taught, or at least it was taught in my day, that comfort is  has when the relative humidity of the premises
does not exceed 65-70%. Upon arrival of  that threshold, a form of ventilation must be activated. The ideal is a hygrometer  connected to an exchanger or, at least, to an exchange opening with  the outside. This is as far as technology is concerned, in practice, the Lady  Maria, she knows for herself when it's time to open a window!

Sure, if it's raining outside and we keep the window open for  an hour, when we close the shutters, we have absorbed a large quantity of water.  Good and good Maria would not behave like that, she knows that for  changing the stale air takes 5 minutes!
If we then cover the internal walls and ceilings with plasterboard, we use the characteristic sponge effect which is considerably higher than that of cement plaster.
The ability of a material to be breathable is expressed in µ, which indicates how many times the material in question is less diffusive  than the air. The value 1 of µ is attributed to air in stationary conditions.  For the speech we made above, that is that the material of  internal finish, plays a temporary sponge effect (between an opening of  window and the other, Mrs. Maria would say), no one would dream of  leave an internal facing with the naked polystyrene which has a µ = 80-200, (a  depending on the type and density) but, for sure, it would place a plasterboard plate with a µ = 8.4 or, better still, a layer of wooden beads  (It is no coincidence that in inhabited attics it is normal to have beads in the intrados  roof)!
All this not to let the wall breathe but to use the internal finish as a lung when the production of water vapor rises, and then hand it back when the relative percentage returns within the comfort limits.
We certainly haven't said everything there was to say about breathability, let's just hope we made a contribution to clarity.

The breathability, the µ value (which can be read mù) and the Sd value
Breathability is expressed with the µ value: the µ value  it tells us how airtight a material is  o vapor permeable!
A low µ value indicates high vapor permeability (very breathable).
That of wood is about 40: that is, forty times more airtight than air.

Examples of µ value:

• in the wood fiber panel it is equal to 7

• in polystyrene it is 70

• in the vapor barrier it is 100,000.

• in the Breathable sheet it is 60,000

• in the coating of the coat 400

In the case of the sheaths, the transpiration power is expressed with

• the Sd value in meters (µ value for the thickness)

• or WDD (in gr / m2 24 h), that is the quantity of water vapor that can transpire one square meter of surface. in 24 hours.

A vapor barrier does not pass even 1 gram of water vapor per square meter in 24 hours. 
The Sd value (i.e. the resistance to the passage of vapor) is obtained by multiplying the  µ value for material thickness:
plaster:  value µ = 5  1.5 cm thick  Sd = 5 x 0.015 = 0.075  hence a Sd = 0.075
drywall:  value µ = 8  1.2 cm thick  Sd =  8 x 0.012 = 0.1  hence a Sd = 0.1
polystyrene:  value µ =  70   1.2 cm thick  Sd =  70 x 0.012 = 0.1  hence a Sd = 0.84
a vapor control sheet or a breathable sheet usually has an Sd value> 5

For example, this is why if we do a  insulation from the inside  It is also a good solution to lay a sheet with vapor control properties: plaster or plasterboard are not really vapor barriers! ​