I answered about the incompleteness theorem in the other thread. Let's talk about some of his other contributions here. (This list is definitely incomplete*, but just some stuff off the top of my head.)
1
The "black hole" theorem (incompleteness theorem) is closely related to, yet subtly different from, the Hawking-Penrose Singularity Theorems. The Hawking Penrose theorems again prove the geodesic incompleteness of spacetime under certain cosmologically reasonable assumptions. The difference is in the interpretation. The Penrose theorem proves the genericity of black hole formation; the Hawking-Penrose Theorem guarantees, in some sense, the genericity of the Big Bang.
2
Penrose made significant contributions to how we understand causal geometry of space-times. A particularly interesting paper is Kronheimer and Penrose, "On the structure of causal spaces" (Proc. Camb. Phil. Soc. (1967)). In this paper they abstracted the relation between two space-time events (as being time like or light like) into a partial order. From this one is naturally led to study the ideals and filters, and their principality. This leads to a beautiful description of what the idealized "boundary at infinity" should look like for space-times.
3
The GHP Calculus (named after the authors Geroch, Held, and Penrose of the 1973 paper "A space-time calculus based on pairs of null directions" (Journal of Mathematical Physics)) and the more general Newman-Penrose formalism ((1962) "An Approach to Gravitational Radiation by a Method of Spin Coefficients" (Journal of Mathematical Physics)) are some of the most common ways to perform symbolic computations in GR.
The calculus is a version of the Cartan formalism (or a special way of looking at Ricci rotation coefficients), but taking special advantage of the four dimensionality of space-time and the Lorentzian structure of spacetime.
4
The Penrose inequality is a conjectured (and partially proven in many special cases) relation between the area of the apparent/event horizon of a black hole space-time with the mass (as observed at infinity) of the corresponding black holes.
This inequality actually lead to a lot of interesting recent works in Riemannian geometry.
5
Also, he formulated and named the Strong and Weak Cosmic Censorship Conjectures.
6
Penrose is also responsible for suggesting his namesake process for extracting energy from rotating black holes through backscattering. The process, combined with some putative nonlinear feedback mechanism, gained popular fascination under the martial name of the Black Hole Bomb. In the literature this is called the superradiant instability and has been proven to work in certain linearized matter models around rotating black holes (such as the Klein-Gordon model for massive scalar waves).
An interesting modern mathematical discovery is that the superradiant instability does not apply to massless scalar fields. Understanding how this works for tensor fields, especially for those solving the linearized Einstein equations, is a massive undertaking and crucial in the current effort to demonstrate nonlinear dynamical stability of the Kerr black hole.
7
Finally, something a bit more whimsical, since I don't know anyone who actually uses it: the Penrose notation for tensor computations. I tried to use it for a few weeks when I was in graduate school, but gave up mostly because they are impossible to type up.
* Pun very much intended.
