Deworming is extremely important because if a dog or cat has parasites, this will interfere with the normal development and it will be an open door for many severe diseases and illnesses that could end up taking the dog's life. Remember that deworming a puppy or kitten should always be indicated by a veterinarian according to the age, weight, if he/she lives with other pets, as well as the coat and skin condition of the dog or cat. It's best to do an exam of the feces for intestinal parasites beforehand to be able to confirm if the dog or cat has parasites, and in the event that he/she does, find out what type to be able to de-worm the puppy or kitten in the most effective and rapid way possible.
Tapeworms: The dog tapeworm (Dipylidium caninum) is a common infestation in domestic dogs and cats. They are made up of segments.

These segments are the shape, color, and look of a cucumber seed (except that they tend to squirm around in the feces and around the anus of the dog or cat). 
As Rover sleeps, tapeworm segments are passed
When a dog or cat nips at, and swallows, an infected flea, it becomes infected. The larval stage tapeworm, ingested in the flea, is liberated, and attaches and grows into an adult tapeworm inside the animal.
Occasionally, human infants and children become infected with the dog tapeworm. This happens through ingesting larvae-containing fleas. A dog tapeworm infestation in a child is usually quite mild and is easy to treat. The associated dog or cat should also be treated, and flea control measures should be undertaken. Considering the large number of domestic dogs and cats, and the relative commonness of worms in pets, the infestation of children with dog tapeworms is, thankfully, relatively uncommon.
Ascarids (Toxocara canis, T. cati) and hookworms (Ancylostoma spp.) are common intestinal parasites of dogs and cats. Not only can ascarids and hookworms cause disease in their respective
hosts, they are also well-known causes of larva migrans syndromes in humans, especially children. While ascarids and hookworms are most commonly diagnosed in puppies and kittens, infections can occur in dogs and cats of all ages. Dogs can also become infected with Baylisascaris procyonis, the common raccoon ascarid, which can cause serious disease in other animals and humans.
Ascarids Puppies are usually born with or acquire ascarid infections early in life. Kittens do not become infected in utero, but like puppies, can acquire ascarids (T. cati) through the queen's milk.

Hookworms Both puppies and kittens acquire hookworm infections (A. caninum, A. braziliense, and A. tubaeforme) through ingestion of or skin penetration by infective larvae, or from infective larvae passed in their dam's milk (A. caninum). Hookworms suck large amounts of blood from their hosts.

Hookworm egg develops into larva, ready to infect host through the skin
Zoonotic Transmission and Human Disease
Humans become infected with ascarids (Toxocara spp., Baylisascaris spp.) through ingestion of infective eggs in the environment. When a human ingests infective eggs, the eggs hatch and release larvae that can migrate anywhere in the body, a condition known as visceral larva migrans. 
Cutaneous Larva Migrans (CLM)
occurs as red, inflamed lesions in
the skin where the larvae of canine hookworms burrow under the skin
Most cases of human ascarid and hookworm infections can be prevented by practicing good personal hygiene, eliminating intestinal parasites from pets through regular deworming, and making potentially contaminated environments, such as unprotected sand boxes, off limits to children. It is also important to clean up pet feces on a regular basis to remove potentially infective eggs before they become disseminated in the environment via rain, insects, or the active migration of the larvae.
Most pet owners do not know that their pets may carry worms capable of infecting people.
Whipworms: Whipworms (Trichuris vulpis, Trichuris campanula) are common in dogs and found throughout the United States. Trichuris serrata is rare in cats.
How are whipworms transmitted and how is whipworm infection diagnosed?
A dog or cat becomes infected by ingesting foo
d or water contaminated with whipworm eggs. Adult worms lay eggs that are passed in the feces. The eggs must remain in the soil for about a month to mature and be capable of causing infection. An infection is diagnosed by finding the eggs in the feces.
Whipworms developing in the soil. Note the characteristic "plugs"
on either end of the egg
Coccidia:

There are many different species of coccidia, but the
most common infections are with coccidia of the genus Isospora (pictured here).
The information presented here pertains to Isospora species
What on Earth are Coccidia?
Coccidia are single celled organisms that infect the intestine. They are microscopic parasites detectable on routine fecal tests in the same way that worms are but coccidia are not worms and they are not visible to the naked eye. Coccidia infection causes a watery diarrhea which is sometimes bloody and can even be a life-threatening problem to an especially young or small pet.
Where do Coccidia Come From?
Oocysts (pronounced o'o-sists), like those shown above, are passed in stool. After they have adequately matured, they become infective to any host (dog or cat) that accidentally swallows them. To be more precise, coccidia come from fecal-contaminated ground. They are swallowed when a pet grooms/licks the dirt off itself. In some cases, sporulated oocysts are swallowed by mice and the host is infected when it eats the mouse. Coccidia infection is especially common in young animals housed in groups (in shelters, rescue areas, kennels, etc.) This is a common parasite and is not necessarily a sign of poor husbandry.
How Are Coccidia Detected?
A routine fecal test is a good idea for any new puppy or kitten whether there are signs of diarrhea or not as youngsters are commonly parasitized. This sort of test is also a good idea for any patient with diarrhea. The above illustration demonstrates coccidia oocysts seen under the microscope in a fecal sample. Coccidia are microscopic and a test such as this is necessary to rule them in. It should be noted that small numbers of coccidia can be hard to detect so just because a fecal sample tests negative, this does not mean that the pet is not infected. Sometimes several fecal tests are performed, especially in a young pet with a refractory diarrhea; parasites may not be evident until later in the course of the condition.
Can People or Other Pets Become Infected?
While there are species of coccidia that can infect people (Toxoplasma and Cryptosporidium, for example), the Isospora species of dogs and cats are not infective to people. Other pets may become infected from exposure to infected fecal matter but it is important to note that this is usually an infection of the young (i.e. the immature immune system tends to let the coccidia infection reach large numbers where the mature immune system probably will not.) In most cases, the infected new puppy or kitten does not infect the resident adult animal.
Giardia: Giardia are actually protozoans (single celled organisms) and are commonly found in the intestines of many animals, including dogs. A few percent of dogs and cats will carry Giardia organisms and not show any signs of disease. Actual diarrhea, bloody or mucousy stool often accompanied by gas production is seen most often in affected puppies and kittens. Giardia can have a significant impact on the health status of malnourished and stressed pups and kittens.
Giardia occurs in two forms: a motile (swimming) feeding stage that lives in the intestine, and a non-motile cyst stage that passes in the feces. Encystment occurs as the parasite travels from the small intestine to the large intestine. The cysts are fairly resistant, and can survive for several months outside the animal as long as sufficient moisture is present. Mature cysts are usually found in the feces of infected animals. Animals become infected by ingesting these cysts. The ingested cysts then break open in the new host's intestine to release the motile feeding stage (trophozoite). Giardia is transmitted from one dog to another through the ingestion of cysts in contaminated feed or drinking water. Cysts may also be found in streams or other water sources. The trophozoite stage may also be infective to dogs. This stage does not survive for very long after being voided in the feces. Therefore, it probably does not contribute significantly to the transmission of Giardia.