About Beth
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As Chris and I drove through Iceland, we saw at least as many horses as we saw sheep. That's a lot of horses.
We went riding twice, both times crossing lava fields near the mountains (the inland portion of Iceland is all mountain). Since they're different from most horses you'll see outside of Iceland, I thought I'd say a little bit about them.
They're short (around 13 hands) and fat, but nobody calls them ponies. They are horses, hestarnir. They just happen to be conveniently within reach of the ground, so you don't need a mounting block to get on them. And if you ride through tall grass, the grass may brush your boots.
As you'd expect from reading about them, the horses were strong, sure-footed, and after a while you start to think of them as the normal size for horses to be (we never saw any tall horses in Iceland, anyway). They scrambled over rocks, up hills, across streams, and over various other terrains that I felt sure we would fall onto. Other things you'll commonly hear about Icelandic horses are that they are good-natured and smooth-gaited. This is mostly true: the first horse I rode was good-natured, and the second was smooth-gaited. :)
In place of the trot, the faster sections of our trail rides were done at a tölt, a gait that one guide explained to me as a four-beat gait similar to a pace, but slower and smoother. It differs from the faster "flying pace" in that the tölt has no period of suspension.
Icelandic horses' saddles are placed further back than I'm used to. I was instructed to put one hand on the biggest lump of muscle on the shoulders, then my other hand next to that, and finally the saddle. Many of the horses wore cruppers to keep the saddle from slipping any more forward than that.
According to the Internet, in 982 the Icelandic parliament (the Alþing, which still exists) passed a law prohibiting horses from being imported into Iceland. This apparently had to do with controlling the spread of disease. Icelandic horses can still be exported, though, and you'll find farms selling and breeding Icelandic horses all over the world (a quick google turns up several in the US and Canada).