As with any other living creature there are a wide range of diseases that fish can get. A full list of these would be nearly impossible;
however this thread will cover a few of the most common symptons. Most diseases can be avoided by keeping your tank clean and doing frequent water changes. If all of the inhabitants of your tank are salt tolerant, putting in the right amount of aquarium salt can also help to prevent disease.

Ich
Description: Ich (Ichthyophthirius) is also known as White Spot. It is a very common parasitic infestation. Many say that ich is caused by rapid changes in water conditions, addition of new fish or poor water quality. The reality is that these events all cause stress to your fish which weakens their immune system and leaves them more open to the disease. Ich is often in your tank unnoticed because your fish are not under enough stress to be affected. Luckily it is as easy to treat as it is common. You must treat your fish swiftly, once you notice the Ich however as it can spread and become lethal very fast.
There are several stages in the Ich parasite life cycle. In one stage they are in your fish, feeding off from them as parasites tend to do. Once they have had their fill, they will drop off into your substrate and form a protective cyst around themselves. There they will grow and multiply. Soon this cyst bursts forth releasing many many more parasites which swim freely in your fish tank until they can find a host. Then the whole process starts over. This entire process can take anywhere from three to seven days. Medication only works in the free floating stage which is why you need to treat your fish for some time after any sign of ich has passed.
Symptoms: Unless you have all white fish, it is easy to spot Ich. Ich appears as small white bumps on your fish's body and/or fins resembling grains of sand. The fish will try to scratch itself on any surface inside the tank as if itching. It may also breath heavy and lose its appetite. Sometimes infected fish will stay away from other fish as they feel weak or exhausted. They may also hover near the surface or other warm areas of your tank.
Treatment: Many commercial treatments are available for ich. Just go to your local fish store and ask. If you have salt tolerant fish, keeping salt in your aquarium also helps both to prevent and to treat Ich. You will have to treat for up to a week after any sign of ich on your fish has passed. This is because the medication only works when the parasites are in their free floating stage in the aquarium.





Body Fungus

Description: Body fungus is just that; fungus on the body of your fish. The fungi feed off from a host fish by releasing chemicals which slowly dissolve the scales or skin of the fish. Poor water quality caused by overfeeding or not cleaning your tank is the most common cause. Body fungus is easy to cure, but must be treated quickly or the fish will die.
Symptoms: White or grey ulcers or patches will develop on your fish, sometimes rimmed with a red line. May look like cloth on your fish. Fish may scratch themselves on any surfaces in the tank.
Treatment: Melafix is a great general antibiotic for aquariums and works wonders on body fungus. Follow the instructions given.




Dropsy

Description: Dropsy is a bacterial infection of your fish's internal organs. Being internal, it is difficult to diagnose and is often not treated early enough to save the host fish. It attacks fish that have been weakened due to excess stress resulting from poor water conditions, overcrowding or being moved. It causes excess body fluids to build up making the fish appear bloated or it's scales to stick out. Early stages of this infection may present themselves as popeye as excess fluid causes the eyes to bulge.
Symptoms: The visible symptoms of drospy come from the excess body fluid buildup inside the fish. Bloating, as well as eyes or scales that protrude more than normal may be signs of dropsy. Avoidance of other fish and loss of appetite are also common.
Treatment: Commercial treatments are available from your local fish store. Just ask as soon as possible. If possible, keep some on hand as you must treat this disease as soon as possible if your fish is to have any chance of survival. Usually by the time the scales of the fish are protruding, it is too late.





Fin or Tail Rot

Description: Fin or tail rot is another type of bacteria similar to body rot. In fact in extreme or untreated cases, the rot will spread to the body of a fish. It generally starts by attacking the thin tail and fins of a fish and may make them shred or appear to be eaten or dissolved. The bacteria often enters tanks through feeder fish or other new fish that are introduced. Most healthy fish will be able to fight off fin or tail rot, and only fish showing symptoms should be treated if possible.
Symptoms: The fins and/or the tail will become frayed, torn or appear to be eaten away at. In some cases, fish will hide or lay on the bottom of the tank away from other fish.
Treatment: General antibiotics such as Melafix will easily cure fin or tail rot. Follow the instructions given.





Flukes

Description: Flukes are parasites that attack the skin and gills of fish. In small numbers they are fairly harmless. In situations where tanks are overcrowded or the water quality is poor, however, they may multiply very quickly and cause harm to already stressed out fish.
Symptoms: Scraping against any surface in the aquarium and heavy breathing are the earliest signs of flukes. At later, more advanced stages, fish will lay on the bottom of the tank with their fins clamped.
Treatment: There are commercial products available at most fish stores for treating flukes. It is very difficult to eradicate them entirely, however, and the best treatment is prevention by keeping a tank clean and not overcrowded.





Hole in the Head - Hexamita

Description: Hole in the head disease is very common among larger fish such as oscars and discus. It must be treated quickly or it will kill your fish. Though it causes ulcers or "holes" in the head and/or body of a fish it is actually an intestinal infection.
Symptoms: As you can probably guess by the name, the typical symptom is "Holes" or eaten away areas on the head or body of fish. The fish may also lose it's appetite and shy away from other fish.
Treatment: There are commercial treatments such as Hex-a-mit available. Ask your local fish store.




Popeye

Description: Popeye is some type of infection or fluid buildup that causes the eyes of a fish to bulge out of their sockets. It can be caused by bacteria and infections may range from minor to fatal. Popeye is often an early warning of dropsy and it is often recommended that you treat it as such just in case.
Symptoms: Eyes of the fish will bulge as if they are going to pop out of the sockets.
Treatment: General antibiotics such as Melafix can be effective, but since it has a wide range of causes, it is hard to treat.




Swim Bladder Disease

Description: A fish's swim bladder is a pouch of air in its body that helps it to swim and float upright as well as providing general balance in the water. Bacterial infections, parasites, and constipation can cause this pouch to become deformed or inflamed affecting the fish's ability to control itself.
Symptoms: Fish may appear off balance or clumsy. It may appear the fish is doing tricks as it will roll around in the tank and have trouble controlling itself. Fish may float or swim upside down.
Treatment: There are commercial products which can be found at fish stores that claim to aid swim bladder infections. Often, feeding your fish a diet of peas for a few days can also help as they act as a laxative, clearing up constipation.




Velvet

Description: Velvet is a parasite which appears similar to and is often confused with ich. The difference is that velvet covers mostly the body of the fish and is a smaller "powder" instead of salt like bumps. Like most similar diseases it is caused by overcrowding, stress, or poor water conditions.
Symptoms: Whitish or yellowish "powder" will form on the body of the fish. The fish may dart around the tank trying to scratch this off as if it itches. Its gills may become clamped and it will start breathing hard.
Treatment: Adding salt (if the fish can tolerate it) can aid in the prevention and recovery of velvet. Commercial chemical treatments are available from pet stores.



Please feel free to add any that has not been mentioned.