Alfred Wegener developed the original theory, but it was very contentious and solid evidence only arrived in the 1950’s and 60’s through the work of several people on sea-floor spreading.
Wegener postulated that all of the continents had once been joined together, and then had been split by catastrophic forces, forming the present continents.
Wegener used the similar outline of the continents, evidence of ice ages that could be connected from continent to continent, the occurrence of the same type of fossils in strata on different continents. and the location of mountains to support his theory.
The theory was rejected because catastrophic forces went against the prevailing theories of uniformitarism. The time line was contradictory to established time lines of Darwinian evolution. The theory of mountain building and sedimentary fossil layers at the time was the geosyncline theory. This well established theory would have had to be abandoned if Continental Drift theory was accepted. Wegener was not a geologists and the idea that there were natural forces that could move continents was incredible.
Sadly Wegener did not live to see his ideas vindicated with the acceptance of the modern theory of Plate Tectonics.
A fuller list of who contributed to acceptance of the theory can be found here: https://en.m.wikipedia.org/wiki/Plate_tectonics under the ‘development of the theory’ section.