Tubifex worms (Tubifex tubifex, most likely).Tubifex are oligochaete ("OLL-ugo-KEET" meaning "few bristled") annelid worms. The species with the universal distribution is Tubifex tubifex, but in Australia the usual aquarium tubificid is a sister genus, Limnodrilus udekemianus. More than four dozen morphologically distinguishable Tubifex species have been discovered, some from pretty specialized environments, like the abyssal sediments of Lake Baikal, the mile-deep rift lake in Siberia. Telling the tubificid genera apart isn't for the amateur:  for one thing, they reabsorb their diagnostic reproductive organs, by which the specialists tell them apart; examining mitochodrial markers, researchers in 2001 determined that there are also morphologically indistinguishable cryptic species of Tubifex
When tubificid species in Malaysian rivers, lakes and artificial ponds were compared with a temperate lake (in Poland) only one species (Limnodrilus hoffmeister) turned up in both tropical and temperate waters. In Malaysian waters, researchers discovered that the tubificids were far more common than the spottily distributed naidid worms, which we may also find in our aquaria; and the lumbriculids, like our popular California blackworms, were absent from Malaya.
Like their distant earthworm cousins, tubificids are hermaphrodites that produce both sperm and eggs in the clitellum, which is that band around the middle. In natural waters, aquatic worms dominate in soft bottom sediments. The richer the water, the greater the population of Tubifex. Thrashing their rear lengths for maximum oxygen, the worms stand on their heads to feed on organics in stagnant lake bottoms, where they build up a protective tube in mud. "Tubifex" means "the tube builder," so don't get side-tracked with even-quite-knowledgeable folks who call them "tubiflex." It's a good error, though: "tubiflex" reminds me of the lax movements of a red rubber garden hose on a warm afternoon.
Keeping tubifex. When you're keeping them, you'll be omitting the mud of course, but don't drown them, as I've done many times. They'll survive better at cool temperatures, to slow their metabolism, and in water so shallow they can reach the surface. I flush them morning and evening, gently, using just enough water force to break up the ball of worms. I've noticed that they survive better if the last rinse is with water that's been passed through charcoal (in a Brita filter). The consensus is, they're stressed by chlorine, but I also suspect copper in my tapwater.
Tubifex fears. Since tubificids can get by in virtually anoxic conditions, their detractors sometimes warn that they ingest anaerobic bacteria, and associate intestinal ailments in fish with feeding tubifex. My own feeling is that anaerobes are likely to predominate in any intestinal fauna, and that includes fish intestines, and the clean conditions in which you've been holding the tubifex have largely purged their intestines anyway. If toxic bacteria are killing fish, dead tubifex are more likely culprits than live ones, to my way of thinking.
Tubifex is also sometimes singled out as the alternative host transmitting the sporideans  that cause "Whirling Disease" in the young of some coldwater salmon and trout species in hatcheries. The specific parasite concerned, Myxobolus cerebralis, lodges in young fishes' brains, as its name implies, and disorders the central nervous system, causing spinning and staggering. This is a coldwater parasite that doesn't even affect  all species of salmonids, let alone any tropical fish. Though other myxosporideans do infest tropical freshwater fishes, tubificid worms haven't been implicated as their transmitters. Other pathogens besides sporideans are vectors of central nervous disorders in our fishes; they include viruses, bacteria and fungi, none of them treatable by the time symptoms appear. If you have lingering doubts, read Ken Wolf and Maria E. Markiw, "Salmonid whirling disease" at the US Fish and Wildlife Service website. Capillaria have been transmitted through tubifex in lab experiments,  but the common method of transmission of Capillaria  in aquaria is from fish directly to another fish.
Tubificid worms can also be intermediate hosts for some tapeworms found in wild-caught fishes, I understand. In the aquarium the tapeworms can't complete their complicated life-histories and aren't ordinarily a problem. Wright Huntley is one who feels that the disease issue can get overblown and can generally be traced to dirty culturing. He avers the innocence of tubifex, according to a post in the thread "Pathogens and Tubifex/Blackworms" at the Live-Foods Mailing List, 13 Dec 1998, noting that tubificids are known intermediate hosts for tapeworms in the wild (where they can complete their complicated life-cycle) but only rarely in aquaria, and from wild-caught fish. A follow-up post from David Robinson the following 12 August is apropos.  A June 2001 e-mail from Barry Cooper was posted at AquariaCentral. It read in part: "Many foods can act as 'carriers' of organisms, particularly bacteria. Thus, tubifex or blackworms, particularly if they are not in good condition (i.e. some are dead or dying) could be contaminated. Similarly, many people believe that improperly stored frozen bloodworms can be a source of bacterial infections. Finally, any rich high protein food can result in a heavy burden of excreted wastes in the tank, which could favor bacterial infection.  I know of no definitive work to prove or disprove a specific role for these worms in disease transmission in aquarium fish, although I do know of some other scientists who do not believe that they act as intermediate hosts for infectious agents. I use blackworms because they are available and are cultured in clean conditions. True tubifex are rarely available these days, although many stores sell blackworms as tubifex. Tubifex, when available, are likely to have been harvested from dirty environments."
This should be reassuring, on the whole. I tend to think the prejudice against tubifex worms may often be connected with these low-life origins, near sewer outlets, as their detractors never fail to point out. Or in the effluent from breweries and distilleries, as Dr. Gunther Sterba noticed. Low and boozy, either way, sort of a "Liza Doolittle Syndrome." In southern California you'll hear that tubifex are brought in from Mexican sewers: a detail, I feel, that speaks as much to Californian prejudice about Mexico as it does to any tubificid reality. Are collectors of wild tubifex still slogging in hip waders through sewage mud to supply your LFS? Even so, after a day or so in clean water, flushed with de-chlorinated tapwater and with their digestive tracts empty or fed with squeezings from your sponge filter— I think, Pickering, that Eliza Tubifex is ready for the Embassy Ball.
As a general rule, the better place to look for pathogenic strains of bacteria is in or on sick fish, rather than in or on invertebrates. Nevertheless, this is not what you'll hear from many authorities who write for hobby magazines or post on the web. Many thoughtful, well-informed fishkeepers have repeatedly gone through episodes of mysterious disease that they have associated with feeding tubifex and which have cleared up when tubifex was eliminated from the program. The evidence for the coincidence is always circumstantial, however.  
Biomagnification. A more urgent concern to my mind, though I never hear it discussed, is the possible problem of "bio-magnification" in any wild-collected bottom-feeders like tubificids. They ingest heavy metals, such as cadmium, in detritus, and store them in body tissue. A predator farther up the food web, such as your fish, concentrates any toxic metals in its tissues and pays the consequences. In a classic example of bio-magnification in the 1960s, DDT from fishes was concentrated in the tissues of fish-eating eagles, which were decimated. Current bio-magnification issues, which deter us from eating Hudson River fish, are centered on PCBs. So, though tubifex don't transmit parasites of tropical fish, I wouldn't feed wild-caught tubifex after all, because of this bio-magnification issue.
At your LFS. Here in New York nowadays, there are no local fish stores selling tubifex, nor, for that matter,  any live food save some overpriced brine shrimp. Everything now  is freeze-dried or frozen.
Culture? Most folks buy tubifex at the LFS, but David Robinson's  post at the live-foods mailing list noted that Dan Carson was culturing them in a tray with garden-store cinders, fed with fish-meal based pelleted fertilizer. Jim Quarles also offered his techniques for culturing parasite-free tubifex. Cultured tubifex avoid the biomagnification issues, and surely they must avoid any lingering possibility of transmitting parasites.