What is the difference between thermodynamic and empirical temperature? When I've studied Thermodynamics I did so in Callen's book and there the author talks about temperature as a single thing, which mathematically is simply defined as:
$$T = \dfrac{\partial U}{\partial S}.$$
Now, currently I'm needing to learn a more conceptual approach to Thermodynamics and in some books I see the authors talk about "two kinds of temperature".
One is called the empirical temperature and the other is called the thermodynamic temperature. As far as I understood the thermodynamic temperature is the one I've always known, which can be defined by the equation I've mentioned.
Now there's this empirical temperature and I have no idea about what it is. The authors introduce it by talking about lots of experiments relating properties of systems to thermodynamic equilibrium. In one of the books it is said:

Let $X$ represent the value of any thermodynamic property such as the emf $\mathcal{E}$ of a thermocouple, the resistance $R$ of a resistance thermometer, or the pressure $P$ of a fixed mass of gas kept at constant volume, and $\theta$ the empirical temperature of the thermometer or of any system with which it is in thermal equilibrium. The ratio of two empirical temperatures $\theta_2$ and $\theta_1$, as determined by a particular thermometer, is defined as equal to the corresponding ratio of the values of $X$:
$$\dfrac{\theta_{2}}{\theta_{1}}=\dfrac{X_2}{X_1}.$$

I don't really get what is going on here. What is this empirical temperature? What is its relation to the usual temperature? And why would anyone define something like that anyway?
 A: The definition of an empirical temperature is basically what the Zeroth Law of Thermodynamics does.
Let us suppose we do not have any prior knowledge about temperature. What we do know is that if we put two bodies in contact with each other they may change some thermodynamic properties -volume, for instance - of one another. When such a thing happens we say the bodies are in thermal contact. After a while the thermodynamic properties stop changing and we say the bodies are in thermal equilibrium. The Zeroth Law consists on the empirical fact that if $A$ is in thermal equilibrium with $B$ and $B$ is in thermal equilibrium with $C$, then $A$ is in thermal equilibrium with $C$. This is an equivalence relation which classify a set of bodies into subsets called equivalence classes. Each class is labeled by a number $T>0$ which we shall call temperature. The Zeroth Law allows us establish thermal equilibrium just in terms of temperature.
To understand the importance of this empirical temperature, defined through the Zeroth Law, imagine a substance - a portion of Mercury would do well - characterized by its volume. You put it in thermal contact with a body $A$, wait for thermal equilibrium and measure its volume $V_1$. Then we freely assign a temperature $T_1$ to this volume and consequently to the body A. If the substance is then put in thermal contact with body $B$ and reaches thermal equilibrium, we measure the volume $V_2$ and arbitrarily assign a temperature $T_2$. The next step is to make use of an interpolation to obtain a function $T(V)$. This is most easily done with a linear interpolation. In this case
$$T=a+bV.$$
In this case
$$\frac{T}{T'}=\frac{a+bV}{a+bV'}\equiv \frac{f(V)}{f(V')}.$$
Therefore this works as a thermometer. 
The drawback of this empirical temperature is that it is not absolute. One can define different scales based on different physical properties, reference points or interpolations. This difficulty is overcame by the thermodynamic temperature which can be defined through a reversible thermal engine operating between two sources. Any of such engine has efficiency 
$$\eta_R=1-\frac{T_2}{T_1},$$
where $T_1$ and $T_2$ are the temperatures of the sources. Given the universality of this result one can for instance arbitrarily define the temperature of the cold source $T_2$, measure - mechanically - the efficiency of the engine and then the temperature $T_1$ is determined by
$$T_1=\frac{T_2}{1-\eta_R}.$$
Note that there is no longer arbitrariness about the concept of temperature, except for the choice the the temperature of the cold source. Therefore it is appropriate to use as reference point which is highly reproducible anywhere. A standard choice is the triple point of water which is defined to be at $273.16\, \mathrm K$.
A: This is a supplement to @Diracology's answer.
If you have a perfectly calibrated thermometer that measures absolute temperature, it will give the same readings as the thermodynamic temperature under most circumstances--that is, an empirical temperature equal to the thermodynamic temperature. However, empirical temperatures are always positive (since they are proportional to volume, voltage, pressure, etc.), whereas thermodynamic temperatures can be negative--even on a Kelvin scale.
How? Let's look at the formula for thermodynamic temperature again:
$$T = \frac{\partial U}{\partial S}.$$
The temperature can only be negative is adding energy to a system decreases its entropy. Seems weird, since adding energy usually increases the random motion of a system's atoms. One exception to this is inside the optical chamber of a laser.
In the simplest case, the active material of a laser is made of atoms with two energy levels: a ground state and an excited state. When the laser is off, almost all the atoms are in the ground state. This is a low entropy state since all the atoms are in the same state. When the laser turns on, energy is pumped in and atoms are elevated to the excited energy state, increasing entropy as atoms are more equally distributed between the states. Here the temperature is positive.
But, once the number of ground state atoms equals the excited atoms, entropy reaches a local maximum, so
$$T = \frac{\partial U > 0}{\partial S = 0} = \textrm{undefined}$$
Odd, to say the least. Pump in more energy, and the atoms begin to concentrate in the excited state, actually lowering entropy.
$$T = \frac{\partial U > 0}{\partial S < 0} < 0.$$
An extremely hot environment has a negative temperature.
This is almost a purely mathematical effect since no instrument can measure a negative temperature.
