Both of the above are correct. I'm just going to elaborate a bit.
Ich is a parasite that is present in almost all water (I believe there are fresh and salt water versions), and is actually present on most fish at some point in their life. One or two ich parasites at a time will not injure a fish. It's kind of like a leech on a human; unpleasant, but a one or two leeches just can't injure us.
The problem comes in when the fish are stressed, as stated above. Another illness, some sort of poisoning (like
nitrate), or even just being kept in a too-small tank can weaken a fish' immune system to the point that ich can flourish.
At that point, ich reproduces pretty rapidly. This is where the "quarantine sick/new fish" comes into play. If introduced in large numbers, ich can overwhelm even a healthy fish' immune system, as Shawnie suggested. So if only one member of a tank is stressed (for whatever reason), and gets ich, the ich can then reproduce to the point that it can actually harm the other fish in the tank.
The final way a fish can get ich is if a tank is treated for ich only until the parasites currently on the fish are dead. This is probably the worst case for any fish. One stage of the parasite lives in cysts in the gravel bed. It takes awhile for these cysts to break open, and when they do, they swarm the fish. Quite a few people here have lost fish to this secondary outbreak of ich. This is why treatment should always go for two weeks or more (two weeks is about the life cycle of the parasite).
