First, the basics: Chrome OS is exactly what it sounds like. It's an operating system that boots directly into Chrome. The OS is a stripped down Debian install, but that doesn't really matter, as we'll see in a bit. Everything happens through the browser window - there's a file browser built into the browser, for instance. The start menu equivalent (of course there's one) is a Chrome logo in the top left corner of the browser. There's no desktop, no My Computer, nothing else - just Chrome.
This brings us to the first major difference between Chrome OS and other OSes. There are no applications to install; everything you could conceivably want as an application is a web application. They make this a bit easier by pinning some shortened tabs ("application tabs", they call them) at the front of the tab list, so that you have one-click access to your Gmail, for instance. Obviously, this is a pretty radical design choice. The emphasis is definitely on shifting to online services for everything, rather than using desktop applications - and, not at all coincidentally, Google has spent years polishing their online versions of desktop applications. (They showed the online version of Office during the demo, but it looked terrible. Half the screen was taken up by the controls. I can't see how it's a serious competitor to Google Docs, in its current incarnation.)
Not only does this moot one of the usual objections to Linux ("I can't run my apps on it"), it dramatically simplifies securing the system. My understanding of it is, Chrome OS has a small number of signed and/or whitelisted programs that it runs, and the system can assume that anything outside that set is a virus. This is such a fundamentally powerful idea that I'm surprised it took this long for somebody to try it out in a consumer OS. Chrome OS then takes it to the next level by signing the kernel and OS, so that there's (probably) no way at all for malicious code to get in. Their goal is for you to be able to trust that if it boots Chrome OS, it's safe to use. As for updates, it automatically updates itself in the background - this is a difficult thing to get right, but Google is more likely than most to pull it off.
Because there are no local applications, they can get away with having no local storage. This bears repeating, because again, this is pretty radical: you can't save files to the hard drive. You can use flash drives and stuff like that, which makes sense, but the OS drive itself is locked down and mounted read-only. This will be one of the more controversial decisions, I'm sure. It forces you to store all your files and settings in the "cloud", which makes migration easier, but is probably going to be kind of a pain.
I'm not totally clear on how it handles user accounts, but my impression is that you'll be able to sign in to any Chrome OS machine using any Google account, and have it work the same. This is an incredibly powerful idea! It essentially means that you're completely independent of the hardware you're using. If your computer explodes, or you spill apple juice on it, or it's stolen by pirates, or whatever - no problem, it's easy to replace. This ties in with the no-local-files thing - if all of your files are already in the cloud, then it makes sense that you'd be able to log into any Chrome OS machine and have it work the same as your own.
A word on the cloud: This is going to be a sticking point for some people. When I say the "cloud", what I really mean is Google's servers - I doubt they'll allow you to switch to other providers, if that's even feasible. There are legitimate user interface reasons for doing it this way, but it still has the potential to be a privacy nightmare, not to mention the power Google is going to have once they hold everybody's user data. Again, this is one of those things that Google can get right if anybody can; the question is whether or not anybody actually can.
Another word on the cloud: When your internet connection is down, or when you're on an airplane, or when Google's servers go down (it's happened before, it'll happen again), your Chrome OS computer is going to be basically useless. People asked about this in the Q&A, and the answers boiled down to "HTML 5 lets you use web apps offline" and "Google has better uptime than most personal computers, so nyah!" It's not really a satisfactory answer, since HTML5 offline storage is only useful for web apps that have been specifically designed to work offline. In the end, Chrome OS won't be that useful if you're someplace without free and easy Internet access.
Random thought: It's accepted among security experts that local access = root access, for a sufficiently determined attacker, for lots of reasons. Google has taken some steps to prevent that (signed OS is a big one), but it remains to be seen whether it's enough. There are a lot of really devious attacks that you can use if you have unfettered local access.
So what's the big idea?
The paradigm that Google is aiming for with this is something called "appliance computing" - treating the computer like any other appliance. You have a basic expectation that a refrigerator, for example, will just work, without requiring constant maintenance. Appliance computing is when computers are that simple to use. This is something that people have wanted for a long time, but the existing model of computation made really difficult. Designing an OS that doesn't require administration is hard, but based on the info I've read so far, it seems like Google might have pulled it off. Designing a truly secure OS is really hard, but while I'm never willing to bet against hackers, I think Google has at least done better than anybody else who's tackled this problem.
Their goal with Chrome OS is netbooks and things like that, which makes sense. While Chrome OS looks nice, I wouldn't ever use it as my primary operating system. There's a certain level of control over my systems that I prefer to have, and one of the key goals of Chrome OS is shifting most of that control (and the associated responsibility) to Google. As a spare system, on the other hand, Chrome OS will be really useful, and that's how most netbooks are used these days anyway.