Spidercat outlined some good points, since the Ichthyophthirius multifiliis parasite is a very popular issue in aquariums i feel like adding my personal experiences as well.
Causes for an outbreak:
This can be but not limited to temperatures, pH and nitrite / nitrate levels, lighting, overpopulation! - environmental changes of all sorts. Anything that causes stress or weakens immune system of the fish.
Avoiding an outbreak:
AVOID STRESS AT ALL COST - when settling new fish, make sure the water temperature and pH levels don't cause a shock - it's highly suggested to measure tank temperature and pH levels and get information from the dealer about the water they use - otherwise measure on your own. If there are big differences use a second tank to bring the fish slowly to the level of your tank.
When settling in new fish I can't underline this enough, use a second tank for 2-4 weeks - specially if the new fish are weak, tiny, unhealthy, - in most cases the "new fish" will get the parasite first, this in most cases is not caused by the fish themselves but by a weakened immune system from the transport from the farm or a shock from the difference of water quality.
High nitrite and nitrate levels can harm the immune system of your fish and accelerate an outbreak.
Again, having a second tank with 30%-100% of the main tank water volume can be an immense help to either quarantine fish or to have "good water" at hand - so you can, if in need, settle fresh water over night and realize 90% water changes (I'll write about this below).
Let's have a look on the life-cycle of the Ichthyophthirius multifiliis parasite.
The white points / dots on the fish are the trophozoites - normally the first trophozoites are spotted on the fins. Once the parasite has gathered enough food he will release from the fish and enter the tomont (dividing) stage. The tomonts gravitate to the ground, but can stick to any ornamental object. The tomont divides up to 10 times - within 7 hours each tomont can divide in 1024 encapsuled theronts which attack the fish - from the the lifecycle repeats itself.
The parasite lifecycle is dependent on the temperature - wikipedia states that at 25°C / 77°F the lifecycle takes 7 days while at 6°C/43°F it takes 8 weeks.
The parasite does not like temperatures above 30°C and pH below 5,6 and above 9,2 - unfortunately, most fish cant stand that temperature and those pH values too - some fish can stand pH values below ~6 which also might be a help.
My fishes got the parasite, what can I do?
First and foremost, analyze the mistakes that lead to the spreading of the parasite - otherwise you will sooner or later face the same problem again.
There is no ultimate solution but a few points that certainly will, could help.
If your fish can withstand it heat up the tank to 30°C - if not, proceed to 2.
Make drastic water changes - 90% are reasonable (Here is why a second tank with prepared water can be helpful - you can resettle the fish every day in the prepared water or just change the water of the tank. AGAIN MAKE SURE THE FISH DON'T HAVE TOO MUCH STRESS.
Get vitamins and Artemia - your fish need to stay strong.
If possible separate the infected fish from the healthy fish (again a second or 3rd tank for this purpose is helpful.
Get an UV-C clearer - in this case bigger / more is better - got multiple filters, put it on each filter, maybe even get one intern UV-C clearer - the more water you can clear from the parasite per hour the better.
AVOID STRESS at all cost - if the tank is in a frequented room put a towel on it.
I did manage to clean fish from the parasite by grinding along wit ha finger (only works if you are really careful, otherwise you do more bad than good)
Clean all plant and other inventory during the sterilization process multiple times.
Clean the gravel during the sterilization process with a gravel cleaner - best to do this before the daily 90% water change - same for the plants and other inventory.
Diatom or sand filter (I haven't tested this yet).
Lower pH (BE VERY CAREFUL HERE UNLESS YOU REALLY KNOW WHAT YOU DO SKIP THIS).
Medication.
Final words:
If there is an outbreak clearly something went wrong. Tanks are a biological environment where every change can put it out of balance.
Fish dealers are not always experts, often they are great salesmen. It's a huge commodity to have a fish dealer who loves aquaristics and is not just doing his job as salesman.
I have been doing aquaristics for 22 years, the best advice I got was online and from books in 90% of times. Reading about water and fish will prevent such problems as ichthyo.
Last but not least, aquaristics is a serious hobby which needs dedication, passion and some financial efforts to have a long lasting positive experience - the reward if those are put in is awesome - I love my hobby :).