The white 'worms' you're seeing on your bichir are likely Macrogyrodactylus polypteri which are (as has already been mentioned) a type of fluke. As the species name implies, these are specialist parasites that are found only on polypterids (there are other species that parasitize other types of fish) and they can cause the death of your fish if not treated.
Medications containing praziquantel (like Jungle Parasite Clear and Hikari Prazi-pro) are usually very effective at getting rid of these little buggers. I generally administer the medication at half-strength but follow the rest of the prescribed treatment regimen closely.
There are many kinds of 'worms' that are commonly found in aquaria.
Planarians are flatworms that thrive on uneaten food or other organic debris in the aquarium substrate. They can often be seen gliding slowly over the aquarium glass. They are harmless to your fish but an overabundance of food will lead to a population explosion of these 'worms' that can be unsightly.
Nematodes, also called roundworms, are also quite common in aquaria. The majority of these are free-living (non-parasitic) varieties that look like wriggling hairs as they actively swim...unlike planarians. Like planarians, free-living nematodes are usually found in the substrate but occasionally venture out to find a new location. Excess build up of fish waste and uneaten food in the substrate can also lead to a population explosion of nematodes.
There are also many types of parasitic nematodes but they are generally internal parasites that you normally only see when they protrude from an infested fish's anus. These are usually effectively treated with praziquantel-containing medications.
-Joe
