What does it mean for a surface to have a total curvature of $4\pi $?
I have seen that both the catenoid and Enneper surface are the only minimal surfaces that have this total curvature, but I don't really understand what significance this has?
Could anyone please explain this?
Well first of all these have total curvature $-4\pi$, not $4\pi$.
The Gauss-Bonnet theorem tells us that the total curvature of a surface is equal to $2\pi$ times the Euler characteristic of the surface. Moreover we know that the Euler characteristic of a connected surface is always an integer that is at most 2.
So in general we might want to ask: What are all connected minimal surfaces without boundary in $\mathbb{R^3}$ of with total curvature $2\pi n$? We see this is the same as asking that the Euler characteristic be n.
Well, if $n=1,2$ the answer is none, since a minimal surface has nonpositive Gaussian curvature everywhere. Now we can go down from there trying to understand the most topologically simple minimal surfaces in $\mathbb{R^3}$.
For $n=0$ it's not hard to figure out that a plane is the only connected minimal surface without boundary in $\mathbb{R^3}$ that has total curvature 0.
What you are saying is that these are the only two minimal surfaces in $\mathbb{R^3}$ with Euler characteristic $-2$. So we see this case still has a relatively nice answer.
But as we decrease $n$ even further the story becomes increasingly complicated, and a general answer is unknown.