The Speed of Light

The approximate speed of light was already known to us back in the time of Isaac Newton. Astronomers were able to use the rotation of planets and their moons as an incredibly precise system of "clockwork", and the precision of these measurements was so exact that they could identify the changes in apparent timings caused by light taking longer to reach us from more distant parts of the solar system. The critical measurement was that of the eclipse of the moons of Jupiter -- Roemer noted that the eclipses were seen slightly earlier when Jupiter was nearer to us, and slightly later when the planet was further away.
Newton's quoted estimates in Opticks of light taking seven or eight minutes to reach us from the Sun, along with an estimated distance of the Sun of seventy million miles, would have given an estimated speed of light of 150,000-160,000 miles per second. More modern values of a bit over eight minutes (~500 seconds) and just over 93 million miles give us a speed of around 186,000 miles per second, so the old figures weren't that far off
James Maxwell's work on electricity and magnetism in the mid-Nineteenth Century then led to a prediction of the existence of electromagnetic waves that just happened to propagate at the same speed as light. Maxwell argued that light was an electromagnetic wave, and that visible light consisted of electromagnetic radiation whose wavelengths happened to be in a suitable range for human eyes to be able to detect it.
Maxwell's work suggested that the speed of light should be constant, but didn't tell us exactly what sort of lightspeed constancy ought to be involved.....

G. F.Fitzgerald and H.A. Lorentz pointed out, around the turn of the Twentieth Century, that if lightspeed was absolutely fixed with respect to a background frame, but observers moving with respect to that background frame contracted (and perhaps time-dilated) in a particular way, then it'd be impossible for them to use round-trip measurements of the speed of light to work out whether they were "moving" or "stationary".
Einstein then took this system of "Lorentzian electrodynamics" and rederived it in more minimal form to produce his special theory of relativity. If we said that light was globally constant for all inertial observers, then the effects associated with Lorentz's "special factor" would absorb the disagreements that we'd otherwise expect between these observers, over whose frame was the "real" frame for the propagation of light. Although it seemed impossible for the same lightbeam to have the same totally-constant speed in everybody's different frames, special relativity's redefinitions of distances and times created a system in which this could work.
If a lightbeam links two agreed events, special relativity says that two differently-moving observers can disagree as to the distance that they believe the lightbeam "really" traveled and the amount of time that it "really" took to do it, but the combination of those two things, modified by the appropriate Lorentz factors, would combine to produce the same nominal value for the speed of the lightbeam for both observers. For this system to work, we need the lightbeam to travel in a simple way that isn't disturbed by the motion of any nearby objects ... we say that the geometry of spacetime, as defined by lightbeams, is "flat" for all observers with simple inertial motion.
When Einstein wanted to extend the principle of relativity to deal with all forms of motion, he immediately ran into a problem. Gravity bends lightbeams, and a lightbeam that seems straight and constant for an inertial observer can appear to mark out a variable-speed curved path for an accelerating observer. So special relativity's concept of lightspeed constancy didn't work in a more ambitious theory that also had to be able to deal with accelerations and gravitational effects. Gravity didn't just appear to alter light-distances, it mangled clockrates too, so for two different observers drifting in deep space in different gravitational environments, their different rates of timeflow could lead them to assign different speeds to the same lightbeam. These effects also cause a lightbeam to take longer to cross a more "gravitationally-dense" region than one in which the background gravitational field intensity is weaker ("Shapiro effect").
Under general relativity, the user can respond to these variations by deciding to define distances and times locally. It's no longer necessary for us to apply the earlier SR idea that lightspeed has to be globally constant across the region, it turns out that Nature is happy to violate that rule, as long as lightspeed is still locally constant. So if an observer is drifting in a strong-gravity region where gravitational time dilation is causing their clocks to run at half the speed that we'd otherwise expect, then the same slowing effect should make light move across the region at half the usual speed as well. Someone far outside the region might argue that light is appearing to cross the region more slowly than usual, but to a local observer, whose local references are warped by the same degree as the propagation of light, the speed of adjacent light seems to be exactly right. If it seems to have a different speed somewhere else, well, that's someone else's problem.
It could now be argued that since we had learnt that only local c-constancy was necessary (and that SR's "law" of the propagation of light wasn't a law after all), perhaps the geometrical basis of the earlier and more restricted"special" theory wasn't valid. Einstein preempted this argument by designing his general theory to reduce to the special theory over small regions of spacetime. He then argued that the special theory wasn't invalidated by general relativity, but instead lived on within it as a limiting case
Towards the end of his life, Einstein wrote that he no longer considered the decision to construct general relativity as a two-stage model, with "curvature" arguments built on top of a flat-spacetime "SR" foundation, as justifiable. It had been the best that could be achieved at the time, but with the benefit of hindsight it didn't deem to be defensible. Quite what Einstein may have meant by this, what the alternative might have been, and what the implications might be of having a general theory that didn't have a forced reduction to special relativity, still seem to be unresolved questions.