ISO used to refer to film speed, but in digital camera terms, ISO refers to another aspect of light sensitivity within your camera. It affects how "fast" the camera reacts and inputs light within the photo.
1) Noise or degradation of the image is the main problem when using a high ISO number. A high ISO, say 1600—3200 will create noise or static fuzz on your photos. This can be very problematic if you ever want to print your photos. If you're just posting images online, this noise is pretty negible.
2) A "sweet spot" ISO can be anywhere between 100-200. If stretching in low light, up to 800 is how high I like to go (only when absolutely necessary!). When I can, I stick to 100, so that my pictures will have the highest quality look, without any degradation or noise.
3) Use the noise! Sometimes, having all that static/degradation can make an photo appear atmospheric, old or sexy. Many old black and white photos have visible specs from a high ISO and this can be very cool. Just be aware of this effect and when it's appropriate.
4) Auto ISO can be good. Especially for web, where the high ISO won't really be noticeable. If you're in a jam and can't remember what's what, AUTO on ISO I think is okay. If you are ever to print these photos, however, these AUTO ISOs can become a hairy issue. Art directors will say, "Why are they so grainy and noisy?" That's a sinking feeling/feedback. So if you're moving to a more professional status, learn your ISO comfort levels, if you're shooting exclusively for web, no worries! AUTO is the way!