"Help! I've noticed tiny white worms in my tank. What are they? Will they kill my fish?"
This sounds like a case of Planaria. Planaria are flatworms, related to flukes and tapeworms. They are quite small, and if examined closely have eyespots as well as protrusions from the sides of their heads.
Although they do not harm fish, they love to feast on eggs, and therefore are dangerous if breeding egglaying fish.
The Real Problem
Even though Planaria do not harm fish, they are an indicator that the aquarium has a problem. They require a food source, which means there must be excess food wastes in the tank to support them.
If you have Planaria it's time to give your aquarium a thorough cleaning. Pay particular attention to the substrate, as that's usually where they find their food. Make sure there are no dead plants, or other wastes in the tank.
You should also take a hard look at how much you are feeding your fish. Usually the excess waste the Planaria are living on is the result of overfeeding your fish. Reduce the amount you feed your fish, as well as the frequency of feedings.
Finally, once you've cleaned things up, be sure to continue with regular maintenance and water changes. A clean tank is the best defense against becoming overrun with Planaria.