Double blind describes any decision process where all parties directly involved are not given crucial information in order to avoid biasing results. It is most commonly used in the scientific method. In double-blind studies, both the experimenter and the subjects do not know which of the subjects are in the experimental or "treatment" group(s) and which are in the "control" group(s).
The essence of the idea is that both the observer/experimenter and the subject are "blind" to what part of the test they are conducting, hence the "double" blind.
It can also refer to other decision-making, such as in peer reviewed journals, where double-blind means that the reviewers will not know the author's names, and vice versa. In practice, due to people in a field knowing each other and each others' work, sometimes this "blindness" can be rather transparent.
The most persuasive type of medical study, the randomized controlled trial, (RCT) is usually double blind; however, it is not always possible to blind someone to an intervention. For instance, if someone is studying chiropractic techniques for managing low back pain, it is difficult for the subject and the experimenter to be blinded regarding the treatment. Sometimes this bias is partially corrected in "cross-over" trials, in which half the subjects receive one intervention, and then "cross-over" to the other intervention. Where it is possible to administer a convincing "placebo", or "mock treatment", as in handing the patients pills, double-blind testing provides the most reliable results.
Few patent medicines have been subjected to double-blind studies, as the FDA does not require any proof of safety or efficacy, and few producers wish to submit their products to such scrutiny.
Medicines undergo statistical tests because the apparent effectiveness or ineffectiveness of a drug in a single person could simply be random coincidence due to external factors, unrelated to the medicine, having an effect on that person's health. Apparent effectiveness can also be due to subjective biases of the patient or the doctor affecting their reporting of symptoms. People often feel better when they think they have been given medicine, even if they haven't.
The first problem is solved by testing on a large group of people, instead of just one person, and taking the average. The chance of a large number of people coincidentally showing the same response is much, much smaller. By also testing a similar group of people who didn't take the drug, you can compare the two groups and isolate the effects of the drug from what is probably due to either the placebo effect, or simple remission.
The second problem is solved by not letting either the doctors who assess the treatment, or the patients themselves, know whether they received the medicine or not. Ironically, this has led to the "nocebo effect".
Double-blind studies are not a magic bullet, and while they are good for removing unintentional bias, the mechanism still does still leave open a few significant flaws.
One problem is that statistical tests can be prone to fraud. Scientific theories must make logical sense in order to be true, and can thus be easily recognized by anyone as true or false. Statistical tests, on the other hand, have no such requirement as it's often the statistical test that determines whether a theory makes sense. So it is possible for researchers to fabricate data without easily being caught. In this case, only further replication of the test and a wider range of studies can catch any fabrication.
While academic misconduct is a severe problem, there is also a known degree of systematic error in statistics. Tests are prone to randomly give a false positive every so often, or perhaps a false negative (depending on the sensitivity and specificity of the test). By convention, tests are accepted at a 95% confidence interval; i.e., there is a 5% chance that the test has worked only by chance, rather than convention. This allows researchers to continually repeat the experiment until they get the false positive, and only publish that study. This is alarmingly common in big pharma studies and is also known as publication bias. This problem can be avoided through trial registration, where intent to study something is registered in advance; so if the results don't show up, people can ask questions as to why.