So, can I use a non-Nikon flash or not?

Your Nikon will trigger many 3rd party flashes, but with varying degrees of sophistication. From a physical standpoint, you can mount many 3rd party flashes on your camera, and the camera will be able to make the flash fire, but the flash may or may not be able to understand other instructions from the camera that would let the camera control the flash power and timing.

If all you want to do is to put the flash in Manual mode, set the power yourself, and have it fire when you trip the shutter, you can do that with a large variety of flashes.

If you want the camera to adjust the flash power with TTL metering, or do some form of high-speed sync (where the flash has to fire several times during the exposure), or use second-curtain sync, or work as a master unit that conveys any of the preceding stuff to off-camera slave flashes, then everything depends on exactly which flash you're talking about.

I can't be completely sure, but it seems to suggest that there's some sort of communication channel between the flash and the camera. That, I presume would only work with a genuine Nikon flash.

You've got the right idea. Some manufacturers such as Yongnuo have reverse-engineered the proprietary systems and built flashes that implement all or most of the features that the camera offers, and using those can offer a more affordable way to take full advantage of the camera's capabilities. There's no guarantee that these 3rd party units will work as well as Nikon's, or that they'll continue to work well with future Nikon bodies, but the lower price may make up for that.

On the other hand, a lot of people use nothing but manual flash anyway, and you can certainly go a long way with nothing but M mode. I'd suggest reading David Hobby's Lighting 101 course at strobist.com and maybe watching some of Zack Arias's videos at dedpxl.com to get a sense of how much you can do with simple manual-mode triggering.

What about extender cables? Do you need a Nikon-specific one?

As Matt Owen points out in the comments, you'll need a Nikon-compatible cable if you want to use all the features of Nikon's TTL system. That doesn't necessarily mean a Nikon-branded cable, just one that's designed to communicate all the signals from the Nikon hot shoe to a compatible flash. If you're just using manual mode, requirements are significantly less.

By the way: Everything I said about Nikon above can also be said about Canon and it's own system.

One word of caution: Some very old flash units use very high trigger voltages (up to several hundred volts!), and these can damage your camera. It sounds like y you haven't bought any flashes yet, so it's unlikely that you'll have to worry about this problem, but if you're thinking about buying used stuff from eBay or digging through your great uncle's stash of photo gear in the attic, you should either measure the sync voltage or check this list before trying that old dinosaur on your spiffy new camera.