Thermodynamic conjugate variables are pairs of variables $x_i$, $X_i$ where the product $X_i\cdot dx_i$ has the dimension of energy and actually appears as a term in the infinitesimal variation of energy, free energy, or work such as $dE$.

In the pair, one quantity is intensive and the other is extensive.

The best example is pressure and volume because $p\,dV$ is a term in $dE$. Analogously, temperature and entropy because of $T\,dS$, chemical potential and particle number because of $\mu \,dN$, and many electric, magnetic, gravitational, and surface tension examples exist.

If you have 
$$ dE = \dots + p\,dV, $$
it's easy to see that $p$ is the partial derivative of $E$ with respect to $V$. This role of $p,V$ may be reverted if you consider $H=E+pV$ instead of $E$ i.e. substitute $E = H-pV$. Then you will get
$$dH = \dots - V\,dp.$$
This "Lagrange duality" may be applied to any pair.

Question about thermodynamic conjugate quantities
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