Most infectious diseases we know and love are caused by (1) bacteria, which are a kind of single cell living organism, in which case standard practice is to take antibiotics (e.g. leprosy, plague, cholera and many common wound infections), (2) viruses, which are a piece of nucleic acid (DNA or RNA) wrapped in a thin coat of protein that replicates itself only within cells of living hosts (e.g. the flu, herpes and AIDS), in which case the goal is to develop vaccines for the worst ones before you get infected, as most virus caused diseases cannot be effectively treated with antiviral drugs once you have them, or (3) parasites, which are multicelled organisms that live inside other organisms (e.g. malaria and Lyme Disease), in which case breaking the life cycle chain of the parasite (e.g. by killing mosquitos that carry malaria) is the usual approach.
There are also many diseases that are commonly believed to be non-infectious in nature, for instance, because they are hereditary (e.g. Down syndrome and hemophilia), or because they are reactions of an otherwise healthy body to adverse conditions like malnutrition (e.g., goiter (iodine deficiency) and scurvy (vitamin C deficency)) or exposure to radiation. There are several other kinds of non-infectious diseases as well, sometimes your body basically just outlives its warranty period and parts of you wear down from ordinary wear and tear, for example, and real life isn't always so simple. For example, some kinds of infectious diseases affect people who have a hereditary disposition to suffer from them from triggered by an infection (multiple sclerosis is likely to be in this category).
Prion diseases are in another category. I'll let someone else explain what they are for you:
As background, remember that prion diseases include mad cow disease, scrapie in sheep, wasting disease of elk and deer, and CJD in humans. The basic problem is the same in all of these--a protein that is a normal part of our nervous system starts converting to an insoluble alternate form of that protein, in essence, starting to misfold into junk rather than a useful protein. This junk not only gums up the works in the nerve cell, but it starts to convert yet more normal protein into the junk form.
Prion disease is generally something that happens VERY slowly. The occasional random misfolding event of a few protein molecues starts to convert normal protein into misfolded junk. But since it starts with only one or a few misfolded versions, it can take decades for there to be enough misfolded junk to create symptoms. One theory is that this happens in all of us, but we die long before it becomes a problem. Some genetic defects contribute to this process by making the normal protein somewhat more likely to misfold. This speeds up the process, but still takes many years to lead to symptoms. Even with a genetic defect contributing to the disease, it is a disease of old age.
The first time prion disease was observed in young humans was in New Guinea where certain funeral rites involved cannibalism. People who participated in such rites tended to get prion disease (in this case referred to as Kuru, but it is much the same as all other prion diseases) at a young age. The transmission of the disease could be linked to a particular funeral, probably identifying the person who first developed it and, in death, spread it to family members during the funeral rites. Once cannibalism was stopped, new transmission of Kuru stopped.
The next time prion diesease was observed in young humans in large numbers was in Britain. At first no one could figure out why a rare disease that generally hits the elderly suddenly was hitting unusually large numbers of young people. Ultimately it was traced to the consumption of beef from cattle that had prion disease (mad cow disease).
In essence these cases where disease spread by consumption of diseased tissue give the body a sudden influx of misfolded protein, which can then start misfolding normal protein at a greatly accelerated rate.
Short version: Prion diseases are caused by screwed up proteins and often affect the nervous system. They are often transmitted by consuming infected brains or nerve tissue. (Slightly more learned versions are found here and at Wikipedia).
Prions are particularly scary because lots of things we do to kill viruses, bacteria and parasites in the course of cooking or sterilizing things, don't work on prions.
The bad news is, that these nasty prion guys can get into urine. Since exposure to urine (or in some cases blood) from a prion diseased animal is more common than eating its brains or otherwise eating large quantities of nerve tissue, this is a serious concern. The good news is that this doesn't appear to be an important source of prion disease in humans (based on epidemiology) -- people have gotten it from eating infected animals, not by getting pissed on by infected animals. But, it could make it much harder to control an outbreak in domesticated animals once some are infected, even when standard precautions (like vegetarian animal feeds) are taken. Thus, you could investigate where your food came from, get entirely satisfactory answers, cook carefully and still end up getting infected with a prion disease from an infected animal.