There are two different ideas that are mixed up in this question. Let me try to separate them.
The first has to do with the form of a thermodynamic phase diagram. Consider, for example, the phase diagram of a magnet in the plane of temperature T and external magnetic field H. This is a 2-dimensional space. As GGphys points out, there is a line in this space across which the magnetization changes discontinuously. For a symmetric magnet, this line is on the $H=0$ axis, from $T=0$ to some higher temperature $T=T_c$. If we cross this line, say from $H >0$ to $H<0$, there is a discontinuity in $M$, and we call this a first-order phase transition. At very high temperature, $M = 0$ when $H = 0$. In practice, on the line of discontinuity, $M$ decreases and goes to zero at $T = T_c$. Above $T_c$, there is no discontinuity. We call the behavior at $T_c$ a second-order phase transition, and we call the point $H=0, T = T_c$ the "critical point".
The vicinity of the point $H=0$, $T=0$ is a special place with long-range correlations of the magnetization and, singularities in thermodynamic functions. For example, the magnetic susceptibility $\partial M/\partial H$ goes to infinity at $H=0$, $T=0$. So a special, more sophisticated, theory is needed to describe this region.
This is all for a 2-dimensional phase diagram. If we add more thermodynamic variables, all of this extends to higher dimensions. An example is an antiferromagnet in an external magnetic field or a liquid-gas system with a varying density of a solute in the liquid. Maybe the example of the antiferromagnet is most clear. We can imagine a "staggered" magnetic field $H_s$ that turns on the antiferromagnetic order. The variables are then $H_s$, a uniform field $H$, and the temperature. This is a 3-dimensional thermodynamic space. The line of discontinuities becomes a plane of discontinuities in the plane $H_s$ = 0, and the second-order phase transition becomes a line of critical points at $H_s = 0$, $T = T_c(H)$. There can also be other features. If you make the uniform $H$ field very large, all of the spins will line up uniformly. So if you start from the antiferromagnetic phase and raise $H$, eventually you will find a transition (usually, discontinuous or first-order) above which there is a uniform magnetization. It is hard to make a staggered magnetic field in the lab, so when we measure the phases of an antiferromagnet we are restricted to the $H_s = 0$ plane. This has a (vertical) line of second-order phase transitions at $T = T_c(H)$ and a (horizontal) line of first-order phase transitions at $H = H_f(T)$ that meet and end at an even more special point called the "tricritical" point. As you add more thermodynamic variables, more complicated behaviors are possible.
So far, I have not talked about Landau theory. Landau theory is an approximation to the free energy in which you assume that $M$ is small and write the Gibbs free energy as a polynomial in $M$. This gives an approximate description in the vicinity of the critical point. Even away from the critical point, Landau theory gives a qualitative description of the phase diagram. For example, for the antiferromagnet above, the Landau theory would have a polynomial in $M$ and the staggered magnetization $M_s$, and it would show the various phase transitions that I have described.
Landau theory is then useful to get a qualitative picture of the phase diagram in however many dimensions you have. It turns out that it is also useful as a starting point for a quantitative theory of the thermodynamics near a critical point.
