He is the father of the fractals and the mathematics of chaos. He offered us these memorable words:
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
He brought mathematical rigor to the messiness of reality without the oversimplification that his predecessors had been forced to resort to and showed us how seeming randomness and immense detail can emerge from simple, deterministic mathematical formulas. The impact of this insight on how we understand the very nature of the universe rivals the insights of Newton (the a small number of natural laws explain most of what we observe in the world), Einstein (that time and space dimensions are themselve malleable in accordance with certain physical laws) and the collective discoverers of quantum mechanics (that the laws of physics are effectively laws of probability rather than being apparently deterministic).
Mandelbrot's insights, for example, explain how slight randomness at an infinitessimal scale can limit the extent to which even massive phenomena like weather can be predicted.
The alternative definition he provided for a "dimension" as he used the concept in the context of "fractal dimensions" continues to motivate theoretical physics in areas like loop quantum gravity, where that definitional strategy can show how a four dimensional world can emergently arise from a system of nodes and connections where no coordinate system is initially defined that could help to explain at a deep level some of the oddest apparent properties of quantum mechanics, like its apparent non-locality, a possibility quickly ruled out by mere simplifyinig assumptions in the real analysis that undergirds conventional mathematical physics.