An illustrated construction
They are diagrammatic representations of spacetimes in General Relativity. On them, radial null geodesics move along 45° lines. Causal curves are represented by lines whose tangent is tilted at most 45° from the vertical axis.
Trade the Cartesian coordinates $(t,x,y,z)$ for spherical ones $(t,r,\theta,\varphi)$:
$$ds^2=-dt^2+dr^2+r^2\,d\Omega^2,$$where $d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$ is the standard metric of the unit 2-sphere. The angles are suppressed here and in the next plots. Every point of $\{r > 0\}$ stands for a 2-sphere of radius $r$.
Make the further change of coordinates
$$\begin{cases} \htmlClass{c-u}{u} = t-r & \text{(retarded time)}\\ \htmlClass{c-v}{v} = t+r & \text{(advanced time)}\end{cases}$$Lines of constant $u$ can be identified with outgoing radial light rays; lines of constant $v$ are ingoing ones. In the $(t,r)$ plane both run at 45°.
In these coordinates the metric becomes
$$\begin{gathered}ds^2=-du\,dv+r^2\,d\Omega^2,\\ r(u,v)=\tfrac12(v-u).\end{gathered}$$$(u,v)$ are null coordinates: their level sets are null cones formed by outgoing/ingoing null geodesics emanating from the centre, like the cone of the event $t=r=0$ highlighted here.
Since the radius cannot be negative, $u$ and $v$ are not independent:
$$r\ge 0\iff v\ge u.$$Drawing the $u$ and $v$ axes tilted by 45°, the physical region is the half-plane to the right of the diagonal $u=v$; the other half is excluded.
Both $u$ and $v$ still range over all of $\mathbb R$, so the spacetime representation still occupies an unbounded set. The trick is to rescale each null coordinate into a bounded interval:
$$\begin{cases}\tilde u=\tanh u\\ \tilde v=\tanh v\end{cases}\iff\begin{cases}u=\operatorname{arctanh}\tilde u\\ v=\operatorname{arctanh}\tilde v\end{cases}$$The lines $\tilde u=$ const and $\tilde v=$ const are the very same light rays of the previous plots.
Keep scrolling. To make the change visible we deform $u\mapsto\tanh(\lambda u)/\lambda$ continuously, from $\lambda=0$ (no change) to $\lambda=1$ ($\tanh$). Light rays keep their 45° slope while infinitely distant regions arrive at a finite distance.
The coordinate change brings the metric into the form
$$\begin{aligned}ds^2=&-\frac{d\tilde u\,d\tilde v}{(1-\tilde u^2)(1-\tilde v^2)}\\&+r^2\,d\Omega^2,\end{aligned}$$where now $r(\tilde u,\tilde v)=\tfrac12(\operatorname{arctanh}\tilde v-\operatorname{arctanh}\tilde u)$ and
$$-1<\tilde u\le\tilde v<1.$$Because $(\tilde u,\tilde v)$ are also null coordinates, it is common to draw their axes tilted by 45°. The square $(-1,1)^2$ becomes a diamond, and the condition $\tilde u\le\tilde v$ keeps its right half, shaded here.
This triangle is the Penrose diagram of Minkowski spacetime. If we let each point stand for a 2-sphere of radius $r(\tilde u,\tilde v)$, the diagram represents the full spacetime manifold. The spheres shrink to points on the edge $r=0$ and grow without bound towards the right corner. Here the radii are rescaled for visualization purposes.
As in the very first plot, light cones are still at 45° everywhere, so causal relations are apparent: any causal curve, such as the worldline of the massive particle drawn here, is represented by a curve whose tangent is at most 45° from the vertical.
Here are some level hypersurfaces $t=$ const. All of them approach the same point $i^{\,0}$ on the boundary of the diagram, called spacelike infinity. This does not represent an actual point on the spacetime, but is actually an ideal point at infinity.
The level hypersurfaces $r=$ const go from the boundary point $i^-$, past timelike infinity, to the boundary point $i^+$, future timelike infinity.
Radial null geodesics start at the null boundary line $\mathscr I^-$, past null infinity, and end at the null boundary line $\mathscr I^+$, future null infinity. An ingoing light ray goes through the centre $r=0$ and comes out as an outgoing one.
These boundary points and lines represent ideal points at infinity: they do not correspond to actual points of Minkowski spacetime. Adding them is what lets the whole asymptotic structure fit in one picture.
Move the cursor over the diagram, or tap on it. You'll see the coordinates of the sphere at that point, the radial light rays through it, and its causal future and past. Sphere radii were rescaled for visualization purposes.