In order to conduct an experiment to test a hypothesis, controls are the most important aspect. Controls are the tests/assays that let you know an experiment is working correctly. In other words, a control lets you know the results of your tests are valid.
Using a trivial example of a standard technique in molecular biology: lets say you want to amplify (make many copies of) some bit of DNA using the polymerase chain reaction (PCR). In PCR, you add DNA to be amplified (copied), primers (short DNA sequences that specify the DNA to be amplified, a buffer containing various salts, dNTP (the building blocks of DNA), and Taq (a DNA polymerase).
After running your PCR, you find that you obtained no product. Why? There is a cornucopia of reasons why...your template (the DNA to be amplified) was no good, your primers are not correct, the dNTPs are bad, the machine that the reaction was run in was not set up correctly, etc. etc. etc.
Now in some cases, such as you are amplifying a known target using standard conditions, the failure is due to some technical difficulty, in other words user-error. Here, controls are often overlooked because these "standard" conditions work so often, people cut corners. The problem with not having controls here is that it takes you longer to trouble shoot the problem. Here, the controls are straightforward: 1. Use primers that "always" work, to test that your template DNA is good; 2. Use your primers on template DNA that is known to work; others (but lets keep it simple with these two). If your test PCR failed but control reaction 1 worked and control reaction 2 failed, then the problem likely lies with your primers....maybe they are at the wrong concentration. If your test PCR failed and control reaction 1 failed, but control reaction 2 worked, then your problem likely lies with your template DNA....maybe it was degraded during purification. If your test PCR failed and control reactions 1 and 2 also failed, then the problem likely lies in your dNTPs, Taq, or machine. The take home message here is that if your experiment fails the controls help you determine what is wrong so you can correct the issue and make progress faster.
In other cases, such as you are testing for the presence of specific DNA in an unknown sample, failure could be due do the absence of the specific DNA (IMPORTANT) or a technical difficulty (TRIVIAL). Why does this matter, lets say you are testing for the presence of a pathogenic organism in a sample using the PCR. A negative result (failure) would lead to the conclusion that said sample was not contaminated with said pathogen. (If you are about to eat or buy for your child said sample, you want to be sure a negative result means the sample is pathogen free, not that the technician who did the study forgot to add Taq). Here controls serve to establish the veracity of some result. Without the control the results are, at least, suspect but to me it means the results are useless.
So controls are important for two reasons. First, they provide technical assistance for day-to-day protocols. Second, they demonstrate that a given result is meaningful.
Does this matter in general for scientists? Absolutely! To take a recent example, in our departmental journal club we discussed a paper that looked at the interaction of a bacterial species with the surface cells (epithelial) of a mouse lung. The pictures presented showed a three bacterial cells next to the host epithelial cells. Based on this, the authors conclude that these bacteria interact with the host cells. However, there are no controls. The lung alveoli are very small spaces, thus there is not much extra room within them. So as a skeptical scientist, I want to know what a non-interactive bacteria would look like. Based on the space available, it seems that by raw chance most bacteria, regardless of interaction, would be in close proximity to the epithelial cells.
This issue of experimental weakness is often dealt with using independent approaches. However, in this manuscript the authors used this "result" as the reason for all their additional studies. Thus, the entire work was dogged by the idea that the "interaction" the authors mention was an artifact (not real).