How does Landau theory describe first order phase transitions? We know first order phase transitions, like second order transitions, also have order parameters. They too can be dealt with Landau theory. 
What kind of temperature dependence of the order parameter (a discontinuous step function?) is needed to explain a discontinuous phase transition? 
How is that functional dependence achieved through Landau's approach to phase transition? 
What is the physical way of understanding such discontinuous change within the arena of Landau theory? Is it some kind of tunnelling that happens between the minima of free energy?
 A: Yes, Landau theory can explain discontinuous phase transitions. As a toy example, consider a free energy of the form
$$F(\phi) = a(T) \phi - b(T) \phi^2 + c(T) \phi^4$$
where $a(T)$ changes sign at $T = T_0$, and both $b$ and $c$ are positive about $T_0$. At $T = T_0$, the two minima of the free energy have the same value. For higher and lower temperatures, which minimum is lower changes. 
This is a first order phase transition, since the equilibrium order parameter $\phi$ changes discontinuously. Note that the higher minimum is still metastable. Physically, thermal fluctuations allow $\phi$ to find the global minimum. Landau theory does not account for such fluctuations, it just postulates they happen.
Such phase transitions are generic when the theory does not have $\mathbb{Z}_2$ symmetry, since the linear term is allowed. If we do have $\mathbb{Z}_2$ symmetry but no other symmetries, then a second order phase transition is instead generic, taking place for
$$F(\phi) = - b(T) \phi^2 + c(T)\phi^4$$
when $b(T)$ changes sign, assuming $c(T)$ positive.
