@MrRoboto: The ISS is in Low Earth Orbit, it's not really in "space". It's similar to throwing a baseball, if you throw it fast enough and there is no atmospheric drag it will either escape Earth's gravity or it will come back to Earth as Earth's gravity attracts it. The ISS goes just fast enough that it doesn't quite escape the Earth's gravity well but not slow enough that it just comes crashing down, it does need the occasional boost though as it is slowly falling back into the Earth.
You can see the effect (which is what Einstein described really) if you have a trampoline or similar object, put a bowling ball in the middle, the trampoline canvas (space-time if you're talking about this in three dimensions) will sag and create a well (gravity well for objects in space-time). Now shoot a marble in the well, shoot it too hard and it will escape the well and go off somewhere, or you'll see it scoots around the object until it "falls" into the bowling ball.
As far as its relative speed, that depends on your frame of reference which is concept from basic physics really. From our frame of reference, it travels at about 8 kilometers per second so it travels around the earth every 1.5 hours.
@Atomant: The ISS is ~240km away from us, it's also pretty big, given you have a direct view, you can indeed see it with a small zoom lens, typical camera zoom lenses can easily enlarge a picture 10-50 times, hundreds to hundreds of thousands of times when you use telescopes, so it's basically (optically) the same as saying "why can I see a thing that's 2.4km away". Same goes for Mars, Saturn and Jupiter, those planets are huge, as in, several times larger than Earth, hence we can see them with the naked eye even. Things near to or beyond Jupiter are a bit harder because they're not that large and relatively dim, hence why we didn't know about them until the Renaissance or later.