No, this is not an indepth discourse on the mathematics behind such diagrams .. I don't claim to be that clever! I am though going to try and outline the characteristics of the three main diagrams used in weather services around the world, and indicate some sites which deal with how they can be used.
There are three diagrams that I am going to discuss in this note:
Thermodynamic (otherwise called adiabatic or aerological) diagrams of various types are in use, and the earliest dates from the late 19th century. They are all, however, based on the same principles, and differences are mainly in appearance. Each chart contains five sets of lines: isobars, isotherms, dry adiabats, pseudoadiabats & saturation moisture lines.
Why do we not use a simple bit of graph paper? Well, although it would be perfectly fine to use such and indeed atmospheric stability etc., could be judged by the shapes plotted on the graph, by using the more rigorous diagrams presented here, calculations based on the basic laws of thermodynamics and humidity can be accomplished very quickly, without the use of calculators etc. The diagrams are such that equal area represents equal energy on any point on the diagram: this simplifies calculation of energy and height variables if required.
It is however, important not to get bogged down in the background, mathematics etc., involved. For basic calculation of such as condensation level, temperature of free convection etc., then it will be enough to remember what the various sets of lines mean, and more importantly, how to recognise them. I have included a little history for those that like these things.
The Tephigram takes its name from the rectangular Cartesian coordinates used: temperature and entropy. Entropy is now usually denoted by capital letter S, but in earlier texts the Greek letter 'phi' was used, hence Tephigram. The diagram was developed by Sir William Napier Shaw, a British meteorologist about 1922 or 1923, and was officially adopted by the International Commission for the Exploration of the Upper Air in 1925. Although this diagram is one of the earliest, it is not the first; that honour appears to belong to the Emagram (see SkewT/Log(P)), used by H Hertz in 1884. Shaw was Director of the UK Meteorological Office from 1905 to 1920, and thus this diagram was introduced readily into that organisation and offices/countries that have been, or are allied to this formation. The tephigram strictly should be arranged so that temperature lies along the x axis, and theta (drybulb potential temperature) lies along the y axis. In earlier texts, you will see this arrangement. However, since about the late 1940s, the diagram has always been used in its 'rotatedright' format, whereby isobars decrease upwards to the top of the chart; rather more logical than the original! (For UKbased meteorologists, this is the diagram that they are most used to, and as such in the uk.sci.weather newsgroup for example, often all thermodynamic diagrams tend to get called (erroneously) tephigrams.) 
The Stüve diagram was developed circa 1927 by G. Stüve and quickly gained widespread acceptance in the United States, and has a simplicity in that it uses straight lines for the three primary variables, pressure, temperature and potential temperature. In doing so though it sacrifices the equalarea requirements (from the original Clapeyron diagram) that are satisfied in the other two diagrams. For practical purposes though, this is not important. 
The SkewT/Log(P) diagram is also in widespread use in North America, and in many services with which the United States (various) weather services have had connections. This is in fact a variation on the original Emagram, which was first devised in 1884 by H. Hertz. In the emagram, the dry adiabats make an angle of about 45degrees with the isobars and isopleths of saturation mixing ratio are almost straight and vertical. In 1947, N. Herlofson proposed a modification to the emagram which allows straight, horizontal isobars, and provides for a large angle between isotherms and dry adiabats, similar to that in the tephigram. This chart has much in common with the tephigram, and superficially at least provides a similarlooking trace when a sounding is plotted on it. Hence the two charts are often confused. 
So, in summary, the chronology of the various diagrams (including a couple not discussed above) is (as far as I can ascertain):
(I would be grateful if anyone who has more information on the history behind the developments of these diagrams would let me know so I can add a little to the above.)
Chart  Date  Who by 
Emagram  1884  H. Hertz 
Tephigram  1922 or 1923  N. Shaw 
Stüve  1927  G. Stüve 
Aerogram  1935  A. Refsdal 
Pastagram  1945  J.C. Bellamy 
SkewT/Log(P)  1947  H. Herlofson 
There are five sets of fixed lines (isopleths) printed / displayed on most thermodynamic charts. They are:
Line  Usual symbol  What they are 
Isotherms  T  Lines of equal temperature 
Isobars  P  Lines of equal atmospheric pressure 
Dry Adiabats  theta  Lines of constant potential temperature 
Saturated (or Wet) Adiabats  theta _{e}  Lines of constant equivalent potential temperature 
Saturated Humidity Mixing Ratio  w _{s}  Lines of constant saturation mixing ratio with respect to water 
In addition, you may find others, such as the MINTRA line on UK Tephigrams; used in the forecasting of condensation trails, and the zero degC, MS20degC and MS40degC isopleths highlighted to pick out significant temperature values.
On another page, I have put a detailed list of how to compare the various types of charts. If you know what you are looking at, then ignore this section.
Sites that explain how to ‘do’ things on these diagrams are rather thin on the ground. However, the following might help:
(CAUTION: web sites of a specialist nature are notorious for appearing and disappearing without warning! Don’t be too surprised to get a ‘404’ error sometimes. You can often find a useful site by using a decent searchengine)
Jack Harrison, a highly knowledgeable pilot based in the UK, has prepared a tutorial which beginners will find most useful.
Unisys site. explanation of SkewT diagrams, indices etc.
http://weather.unisys.com/upper_air/skew/details.html
and of course, you will want to put all this knowledge to the test, so I have put a few links to sites with current data below:
University of Wyoming (European data)
Barcelona University
University of Cologne (choose 'European Radiosoundings')
The three plots below represent the main thermodynamic diagrams in use around the world.
SKEWT/LOG(P) 
STÜVE 
TEPHIGRAM 
they are intended to demonstrate the differences in appearance between the three versions, rather than be a strictly accurate representation of that particular diagram. For this reason, I have deliberately left off the labels!
This table will attempt to explain the differences between each diagram, and also explain a little about each printed 'line' on the charts, with the units usually used for the variable displayed.
Line on diagram 
units used (usually) 
SKEWT/LOG(P) 
STÜVE 
TEPHIGRAM 
degC 
straight & parallel; angled 45deg/slope to right. Equal spacing (linear); angled circa 90deg to Dry Adiabats up to 300hPa. 
straight, parallel & vertical. Equal spacing (i.e. linear). 
straight & parallel; angled 45deg/slope to right. Equal spacing (linear); angled exactly 90deg to Dry Adiabats whole diagram. 

hPa (or millibars in old money) 
straight, parallel & horizontal; increased spacing per unit pressure change with altitude. 
straight, parallel & horizontal; increased spacing per unit pressure change with altitude. 
very slightly curved upwards (i.e. convex towards top of diagram). [ not really noticeable for routine use.]; quasihorizontal; increased spacing per unit pressure change with altitude. 

DRY ADIABATS (lines of constant potential temperature for a dry air sample [ i.e. an unsaturated air parcel path. ] ) 
degC (but strictly, and sometimes found, degrees Kelvin are used) 
curved: approx. 45deg to left near 1000 hPa, decreasing to within 10deg of vertical near 100 hPa. 
straight  sharply angled to left  gently convergent to left. (meet at a theoretical point where P=0; T(K)=0) 
straight & parallel; angled 45deg/slope to left; Equal spacing (linear); angled exactly 90deg to Isotherms whole diagram. 
SATURATED ADIABATS (lines of equivalent potential temperature for a saturated [or 'wet'] air parcel path.) 
degC (but strictly, and sometimes found, degrees Kelvin are used) 
curved  but not constant; on righthand side of diagram, curve starts right and bears left above 400 hPa; on lefthand side, curve starts left and quickly become parallel with Dry Adiabats. 
slightly curved to left with height  curve minimal lefthand side of diagram; a gently increasing left turn on right hand side. 
The only notably curved lines on this diagram: On the righthand side, starts slightly right before curving left; on lefthand end, curve all to left. On most diagrams, not shown above about 50degC. 
SATURATED HUMIDITY MIXING RATIO (lines of constant saturation mixing ratio with respect to a plane water surface.) 
g/kg ( i.e. ratio of mass of water vapour in given volume to the mass of the dry air in that sample. ) 
quasistraight*; angled to right, at less than 45deg to the vertical; gently convergent to a point well above the top of the diagram. (* for practical work can be regarded as straight & parallel) 
quasistraight*; angled to left, at less than 20deg to the vertical; gently convergent to a point well above the top of the diagram. (* for practical work can be regarded as straight & parallel) 
quasistraight*; angled to right at less than 45deg to vertical, i.e. less slope than isotherm. (* for practical work can be regarded as straight & parallel) 
… and diagrammatically, I have attempted to highlight the various lines here…. the diagrams are not true representations of each style or necessarily to scale, but are close enough to pick out the major differences listed above:
SkewT/LogP 
Stüve 
Tephigram 
SkewT/LogP 
Stüve 
Tephigram 
SkewT/LogP 
Stüve 
Tephigram 
SkewT/LogP 
Stüve 
Tephigram 
SkewT/LogP 
Stüve 
Tephigram 
Some definitions/abbreviations: (not already given elsewhere by hyperlink)
degC 
degrees Celsius 
describes the 'parcel path' on a thermodynamic diagram when that parcel is unsaturated: i.e. 'dry'. Taken to be 3degC per 1000 ft, or 9.8 degC per km. 

the ratio of the amount of heat absorbed by an object in undergoing a reversible thermodynamic process to the absolute temperature of the object (dQ/T) is defined to be the increase in entropy. 

g/kg 
grams (of water vapour) per kilogram (of dry air) 
hPa 
hecto Pascal … same as millibars. 
is defined as the temperature an air parcel would have, if it were moved vertically (upwards >> decreasing pressure >> expansion; downwards >> increasing pressure >> compression ), from its existing pressure and temperature to a standard level (usually defined to be 1000 hPa). Provided the parcel remains 'dry' or more strictly unsaturated, then the rate of cooling (upward motion), or warming (downward motion) occurs at the Dry Adiabatic Lapse Rate (DALR), which is 9.8 degC per km ( or 3 degC per 1000 feet ). This value is constant. Once a parcel becomes saturated (i.e., the initially unsaturated parcel is lifted until it cools to its dewpoint temperature), then the subsequent release of latent heat of vaporization offsets the cooling rate, and the Saturated Adiabatic Lapse Rate (SALR) is consequently less than the DALR. (see Saturated Adiabats) 

describes the 'parcel path' on a thermodynamic diagram when that parcel is saturated: Taken to be, very roughly, 1.5degC per 1000 ft, or 5 degC per km IN THE LOWEST FEW HUNDREDS OF MILLIBARS OF THE TROPOSPHERE. Towards a temperature of minus 50degC, tends to the Dry Adiabatic Lapse Rate, as air at lower temperatures holds less and less moisture, hence less offset from the latent heat release. 
Very much in the eye of the beholder here. The tephigram is regarded as nearperfect for strict thermodynamic calculations, and its large angle between isotherms and dry adiabats renders it the most effective as assessing degrees of stability. However, the skewT also has this property, and tephigrams aren't exactly plentiful on the net. The skewT though has curved dry adiabats, though for short vertical calculation perhaps not a major problem. Indeed, for a parcel undergoing saturation, you are going to have to cross over to the saturated adiabats anyway, which on all the diagrams are curved. The Stüve is clean and simple in that it has the most straight lines, and is perhaps intuitive in that isotherms are vertical against a pressure plot. The disadvantage is that the angle between isotherm and dry adiabat is not as great as the other two versions. Again, constant use would offset this minor difficulty. For those used to the tephigram, the skewT is most like what you are used to.