Phase transitions are driven by the minimisation of the free enthalpy of the system: If at a certain
temperature the entropy contribution of the Gibbs enthalpy outweighs the enthalpy contribution in
$$\Delta G=\Delta HT\Delta S\qquad,$$
the hightemperature phase will become the thermodynamically stable one. The precise nature of this
change, i.e. how smoothly or abruptly it occurs, is different for different types of phase
transition. To describe this, phase transitions are classified into
firstorder
and
secondorder
transitions. The order referred to here is the order of the differential of the Gibbs enthalpy
for which a step is observed at the phase transition.
Note about semantics: Unfortunately, the term order is used for two different concepts in relationship to phase transitions. On the one hand, each phase transition involves an ordered (lowtemperature) and a disordered (hightemperature) phase  on the other hand, the order of the transition (in the mathematical sense of the word) determines the severity of the changes as described above.
1st order  2nd order  

The
free enthalpy, $G$,
As a result, there is a
1st order:
kink 

From the discontinuous or continuous behaviour of the free enthalpy, respectively, follows the
shape of the
1st derivatives of $G$,
For firstorder transitions, the kink in $G$ corresponds to a
1st order:
step 

Neither free enthalpy nor entropy are easy to measure, but the
2nd derivatives of $G$,
A step in a function causes its derivative to have a
1st order:
singularity: 



The diagrams above show the free enthalpy and its derivatives as a function of temperature, assuming
that the phase transition is triggered by a change in temperature. The kinks, steps and singularities
are observed
whatever state variable
is being varied, although the direction and steepness
of the slopes of the various functions may be different. For example, we know that the hightemperature
phase usually corresponds to the lowpressure phase at a phase boundary; therefore, the slope of $G$
must be positive if plotted against pressure. However, there will still be the familiar crossover of the
free enthalpy curves for both phases at the transition point.
Mathematically, the distinction between first and secondorder phase transitions is very clear: either there is a latent heat at the transition or there isn't; either the heat capacity becomes infinite at the transition or it doesn't.
However, in practical terms the distinction is less clear. Heat capacities (or other quantities)
cannot be measured continuously because we cannot vary the temperature (or other state variables)
with infinite
precision,
and also because the system needs to be kept in
thermodynamic equilibrium
throughout the experiment, which would require an infinitely slow (quasistatic) experiment. If
all experimental date we have are the blue dots shown in the diagram, we wouldn't be able to distinguish
a firstorder from a secondorder transition at that level of precision.
To complicate matters, an increase of the heat capacity to effectively infinity is sometimes observed in
crystallographic phase transitions but the rise begins gradually significantly below the transition
temperature. This is at odds with the mathematical definition of a singularity as a single point at
which a function has no finite value. Such phase transitions are referred to as
lambda transitions
because the shape of the heat capacity curve resembles the letter λ.
Landau theory
enables us to describe and track the changes of a system as it approaches a phase transition in a
generalised fashion, irrespective of what kind of transition (states of matter, magnetic, crystallographic...)
it is.