One century ago, in January 1912, Alfred Wegener presented for the first time his theory of "continental drift" at a meeting held in Frankfurt, in Germany (*). Wegener had collected geological and paleontological data that gave weight to an old observation: that the margin of continents on the opposite sides of the Atlantic Ocean seemed to correspond; as in a gigantic puzzle. Accordingly, Wegener proposed that the present continents were once joined together but had slowly drifted away from each other of hundred of millions of years. (image source: J. Floor Anthony)
The story of the theory of continental drift spans several decades. Initially rejected by a majority of geologists, it gradually gained acceptance, until it became standard in the 1950s. Later on, it became part of what we call today "plate tectonics" which is a pillar of everything that we know in Earth systems science.
Occasionally, the troubled story of Alfred Wegener's theory has been perversely appropriated by climate deniers to claim that they are discriminated by the scientific establishment. But that only shows that climate deniers don't understand how science works. All new scientific theories are subjected to close scrutiny and Wegener's one was no exception. Its acceptance took time for various reasons, including the start of the First World War, shortly after it had been presented. Mainly, however, it was because at the time of Wegener there was no evidence that continents could actually move and no proof that they actually did. When satisfactory experimental evidence on these points became available, the theory was universally accepted. It is true that the debate on continental drift was harsher than usual, but it was not different than any scientific debate, as you can read in detail at this link. Wegener himself would be appalled today if he could see his name associated to junk science, as sometimes it is (see here, for instance).
The relevance of Wegener's idea of continental drift (and of its underlying mechanism: plate tectonics) is not just related to an old scientific debate. It is the basis of the modern science of Earth's systems, which includes climate science. Continental drift is a manifestation of the dynamic forces existing inside the Earth, in the region that we call "mantle". It is because of the flow of matter from the crust to the mantle and back that the system maintains a concentration of carbon dioxide in the atmosphere sufficient to maintain plant photosynthesis. Without plate tectonics, there could be no life on Earth. Indeed, Venus and Mars have no active plate tectonics and - as far as we know - no organic life.
But plate tectonics does not just maintain some carbon dioxide in the atmosphere. It also regulates its concentration and, with it, the surface temperature of Earth. Carbon dioxide is a greenhouse gas acting as the Earth's "thermostat knob." The mechanisms of plate tectonics have slowly reduced its concentration in order to maintain an average constant temperature, despite the gradual increase of solar irradiation over geological times (some 10% each billion years). This regulation is far from being perfect: during the past Aeons, the Earth saw ice ages and very hot periods but, on the average, temperatures remained within the bounds necessary for life to exist. Unfortunately, this regulation mechanism is too slow to remedy to the perturbation we are causing today to climate by our carbon dioxide emissions. Still, what we know about the mechanism of plate tectonics and its consequences on the past history of our planet should make us more careful about what we are doing now to the ecosystems. This knowledge goes back, ultimately, to the work of Alfred Wegener: scientist and pioneer of Earth systems science.
You can read how modern Wegener's view of Earth systems science was from this excerpt from UCMP (University of California Museum of Technology)
"Scientists still do not appear to understand sufficiently that all earth sciences must contribute evidence toward unveiling the state of our planet in earlier times, and that the truth of the matter can only be reached by combing all this evidence. . . It is only by combing the information furnished by all the earth sciences that we can hope to determine 'truth' here, that is to say, to find the picture that sets out all the known facts in the best arrangement and that therefore has the highest degree of probability. Further, we have to be prepared always for the possibility that each new discovery, no matter what science furnishes it, may modify the conclusions we draw."
* Wegener, Alfred (1912). "Die Herausbildung der Grossformen der Erdrinde (Kontinente und Ozeane), auf geophysikalischer Grundlage" (in German). Petermanns Geographische Mitteilungen 63: 185–195, 253–256, 305–309. Presented at the annual meeting of the German Geological Society, Frankfurt am Main (January 6, 1912).