Let me itemize what I think I understand better about relativity now that I’ve read Bertrand Russell’s book:
 One goal of the theory of relativity is to express the laws of physics in ways that are less obviously tied to the position of the observer. In this way, the point is not to establish that “everything is relative”, but indeed to find a standpoint from which objective facts about the universe may be inferred.

Measuring things in feet rather than meters is one of those obvious differences that shouldn’t matter at all in an objective theory of the world. We should be able to measure things in a “humanindependent” way; our laws of physics should remain unchanged if the coordinates change. If we choose to express coordinates in polar form or rectangular form, who cares? If we choose to make the origin of the coordinate system the earth or make the origin Mars, that choice is obviously a fact about us, not a fact about nature. Devising a system of laws that removes these obviously human traits is what the tensor calculus is for.

I think I understand what the MichelsonMorley experiment showed. If there really is such a thing as the æther, and if that æther has any mass, then we’d expect light to take longer to travel in certain directions than in others. I gather that the MM experiment found no evidence that light travels at different speeds in different directions. Hence, if the æther does exist, it must have no mass … but now that I write it out, why is a massless æther a problem?
(Forgive my spelling it ‘æther’; I just find that spelling so charmingly British.)
Here’s what I still don’t really get:
 I still don’t know what a tensor is, really. So I get the abstract outline of the idea of the problem we want to solve, and the name of the mathematical tool that solves it, but I don’t know the actual math.

Russell makes clear — as every other popular exposition of relativity theory that I’ve read has made clear — that relativity theory didn’t have any data to support it for at least a few years. Russell emphasizes that relativity theory would still need to be confronted — as a logical argument, rather than as an empirical one — even if we had no data to support it. I don’t entirely understand the parts that “should have been obvious” even to Newtonian physicists. Certain points about reference frames make intuitive sense — e.g., that Bob dropping a ball while standing on a moving train will perceive the dropped ball differently than will Jim standing outside the train watching “Bob, train, and ball” as an ensemble; Jim will perceive a ball moving in an arc whose forward velocity matches that of the train, whereas Bob will perceive a ball with no forward velocity at all. Someone standing on Jupiter watching “Jim, Bob, train, ball, and earth” as an ensemble will see it altogether differently: now this larger ensemble is moving around the Sun at a certain velocity, moving away from or toward Jupiter, etc.
…Is that all there is to it? That certain quantities can only logically be described in the context of a given reference frame?

It’s a postulate of the system that the speed of light is constant in all reference frames. I don’t really know why we’d assume that. Somethingsomething Clerk Maxwell. Also, possibly, the MM experiment?

Russell tries, I gather, to construct a vocabulary for what now can be described objectively — i.e., for those concepts that survive even after we realize that much varies with the reference frame. He defines things called ‘events’ and ‘intervals’, but I don’t really understand what these are. The Wikipedia doesn’t clarify these concepts, at least for this reader.

I understand, formally, the Lorentz contraction. It would appear that if I’m traveling the speed of light, and you’re observing me in relative motion, then you would see me as an object of length zero. As I say, I understand the equation formally, but I don’t understand why this should be so. Likewise, I understand the idea that if I’m in a spaceship traveling at the speed of light, and you’re back on earth, and we both carry clocks that were identical when we were both on earth, then my clock will show that infinitely much time has passed for every second that passes on yours — using the same Lorentz contraction, this time in the form of time dilation.
If the principles of relativity theory should apply in a Newtonian world as well — if it’s a logical argument that Newton himself would have needed to confront even in a world without sciencefiction spaceships — then how would time dilation and Lorentz contraction have affected the models that Newton himself advanced? Put another way: would Newton have dismissed all of this as irrelevant for objects traveling at terrestrial velocities? Or would he have seen the difficulties introduced by separate reference frames and reworked his entire conception of space and time?

I understand that large objects in some sense distort the space around them. I understand, somewhat, the elegance of postulating that light travels in straight lines, and that it’s the space rather than the light that’s changing its path when it moves past a heavy object. But I don’t really understand what’s happening when space is distorted. In fact I don’t even know if “space is distorted” is a sensible way of expressing this.

I understand that in some sense referring to “space and time” as four separate orthogonal axes is out of date, and that now one refers to “spacetime” as a single unified thing. In some sense I guess this means the dimensions are interdependent: where you are in space dictates where you are in time. But I don’t really understand this clearly.
It’s my own personal mode of comprehension that’s at fault here. Partly, I think Russell’s exposition would be a lot clearer if he allowed himself some mathematics. As it is, I think he’s in a partlymetaphorical / grudginglymathematical world, and to me it’s not a very clear world. (Similarly: Krugman tries in one of his books to explain power laws as elucidated by Herbert Simon, and to my eye Krugman only made the situation less confusing.)
I need to read more about relativity — more Wheeler, say, and less Hawking. (Real talk: has there ever been a morediscussed and lessunderstood book than Hawking’s?) Moremathematical relativity goes on the queue.