The "speed of the aperture" is not an accurate term. A lens' aperture is simply the opening of the iris. It's primary characteristic is how wide the opening is. There is no speed associated with it. Although on any particular shot, there will be a certain "speed" (shutterspeed) related to how wide the aperture is, it is a term more directly related to the *shutter* than the aperture or ISO.
ISO, in the photographic sense, is a standard measure of light sensitivity for your imaging medium. There are actually TWO standards to speak of (and before ISO became *the* standard, there were *other* standards, like ASA...but let's not get sidetracked). Folks will always refer to "film speed," which is a reference to the ISO standard for light sensitivity in *film* imaging, which technically not the right definition to refer to anymore. Actually these days, most people refer to *digital imaging* ISO, which is similar, but different than film ISO. The term itself--ISO--It's really a shortened reference to the International Organization for Standardization's, which documents many standards, not just film speed and digital still camera exposure index standards.
ISO earned the name "film speed" because a film emulsion that is more sensitive to light would require less exposure time to capture an image (or alternatively, a smaller aperture). A film that took less time to expose a shot was considered a "faster" film, hence, a higher ISO rating. Faster films were useful to shorten the exposure time necessary to catch low-light shots. But a downside to those film stock was that the grain of the emulsion was rather noticable in development--literally, bigger fatter crystals vs. slower film emulsions. You would get grainy pictures. If you ever shot an old b&W 3200ISO film, you would remember how grainy the images were compared to the same image shot with an ISO100 film.
The digital photography equivalent ISO standard is actually an entirely separate set of ISO standards. But it serves the same purpose--a method for assigning and identifying ISO standard speed ratings, ISO standard output sensitivity values, and recommended exposure index values, for digital still cameras. The ISO rating says that a given photosensor array, must be able to produce a particular image quality for a manufacturer to be able to say that it takes images at that ISO rating (technically called a "digital still camera exposure index").
Unlike film, which is capable of working at only one ISO rating, photosensors are capable of operating at various ISO sensitivity ratings because their output signal can be electronically amplified. Is your sensor actually "becoming more sensitive to light?" No. You can't change a sensor's native sensitivity. You're simply turning up it's output signal. So, when you set a digital camera to take pictures at an ISO1600 setting instead of ISO100, you're telling it to amp up the weaker signal so that it will still produce an image of a particular light intensity given a specific amount of incoming light.
But here's the key for you...just like film speed ISO, photosensor ISO loses image quality when you force the photosensors to work at at higher sensitivities and with less light. The root cause isn't emulsion grain (as it is in film ISO's problems), but signal noise. Since you have less light to work with, the camera tries to "turn up the volume" to reach higher ISO sensativities, as if you were trying to hear a very quiet song over your radio. The "image signal" becomes more and more difficult to distinguish from the "static," or noise. You'll see mottled variations in the color of a blue sky, for example, when in real life, the sky is a smooth blue color. That's image noise, and it's significantly more noticable in higher ISO settings.
So, when taking pics, when should you put it in high ISO? The answer is debatable. But if photodocumenting what you see is your objective, the answer is as seldom as you possibly can. Keep the ISO as low as possible to avoid unnecessary noise. This, of course, is only an issue when you're trying for low-light shots without flash. You'll find that in brightly lit scenes, or when using a flash, you can always stick to ISO100 or ISO80 (some cameras go as low as ISO 50). But when the lights dim, and you have an artistic preference for not using flash, that's when you'll have to start pushing the ISO up. What happens is that the shutter speeds start to get so long that you can't keep the subject from blurring in the picture. That's when you know that you're going to have to push the ISO up. If your subject is inanimate (doesn't move), then you can avoid pushing ISO up by putting your camera on a tripod, and use those longer shutterspeeds to get more light in the camera. But if your subject is moving, then you'll have no alternative but 1) push ISO up, or 2) use a flash. If you use a flash, you'll change the artistic impression of the shot. If you push the ISO up, you'll concede image quality. Both are a mixed bag. One thing you could do if you had the option, is switch to a camera/lens that is capable of wider aperture settings, letting in more light. Lenses that can stop down to f2.0 or less are highly sought after for it's ability to catch images in low light without going up to a noisy ISO.
There is one camera company (currently) that is changing the relationship between noise and higher ISO. To get to the higher ISO indexes, instead of just turning up the volume on a regular sensor, Fuji has put the work into developing a more sensitive sensor. Somebody smart is working at that company. They've been blowing away competition with their Super CCD sensor, by getting lower noise in high ISO shots without the need of noise-reduction algorithms (which can reduce noise only by sacrificing image detail), and as far as sensor technology goes, theirs is the best sensor out there to date. Personally, I hope to see other manufacturers follow its lead.
One last note regarding noise, and photography with high-ISO. It can be argued that *some* noise is not only an acceptable part of imaging, but can lend character that can add to an image's artistic qualities rather than detract. Would old world-war II era photographs posess quite as much nostalgia if they didn't have that grainy texture? That sepia tone? That point is of course debatable ad nauseum, when imaging steps away from photodocumentation, and into photography as an art form.