"Steampunk" fiction offers us a glimpse, I think. I believe the biggest "hit" would be to our communications infrastructure; every satifactory form of long-distance communication relies on electricty or electromagnetic forces. (Who wants to depend on carrier pigeons and semaphore stations?) Our knowledge of physics would be locked in the 1860s, which would also have a dampening effect on chemistry. (How can you discover electrons if there's no electricity?)
I posit that land and naval warfare would advance little further than WWI-era technology, and air warfare would be unlikely to progress even this far as it would consits of LTA craft, perhaps including a few zeppelin-launched gliders. Assuming certain advances in steam-power (reliability and miniaturization), the year 2000 might bear a superficial resemblance to 1940 or 1955.
Interestingly, internal combustion would eventually be perfected using non-electrical ignition methods. There are certainly plenty of precedents, such as diesel engines, and this even allows for jet engines. So you are left with a sort of retro world which badly needs long-distance communications, and whose scientific progress is crippled if not outright retarded.
Can anyone imagine a way around the communications problem? Grand military tactics would be stuck in the age of Napoleon, and aircraft would be a dicey proposition at best. Go on, try and operate an airfield without radio- I just dare you.

It's such an interesting concept... I remember a short story in some sci-fi anthology ("The Waverlies", 1945, by Fredrick Brown) about Earth being de-electrified by an invasion of electron-based lifeforms who simply "ate" every electron we could generate; dynamos stopped cranking out power because they were being drained by these "waverlies" and everyone reverted back to horse-power. (Brown overlooked the possibility of non-electric ignition for combustion, it seems.) Of course, that society had already enjoyed the benefits of knowing what an electron was. They had already made the fundamental scientific discoveries.
That's what really puts a knot in my brain; without electron theory, there is no relativity; no quantum physics; no Maxwell equations... The ramifications are staggering.

I mean... If you can't ever really get chemistry without understanding particles, that impacts so many other fields as to boggle the mind. What about magnetism? Without magnetism there are no compasses. How would sea trade progress without it?

Thanks for an interesting thread, yimmy!

Implicitly understood. I ask the question because it falls in a gray area; does this hypothetical world understand any basic principals of magnetism? If not, they might be stuck with using lodestones for compasses, an ungainly option at best. The alternative is to allow the creation of magnets but no understanding of the function. A minor issue, really; just a conceptual issue. As far as I can tell there have been several stories and books which considered how a society would change if electricity were taken away, but none which considered the angle we're on now. I wish I had a physics major around to bounce ideas off of. Or even a chemist or an engineer.

I just found that Fred Brown story I mentioned- turns out I was a letter off- it's Waveries, not Waverlies. And it's online
here.