Negative controls are groups where no phenomenon is expected. They ensure that there is no effect when there should be no effect. To continue with the example of drug testing, a negative control is a group that has not been administered the drug. We would say that the control group should show a negative or null effect.
If the treatment group and the negative control both produce a negative result, it can be inferred that the treatment had no effect. If the treatment group and the negative control both produce a positive result, it can be inferred that a confounding variable acted on the experiment, and the positive results are likely not due to the treatment.
Positive controls are groups where a phenomenon is expected. That is, they ensure that there is an effect when there should be an effect, by using an experimental treatment that is already known to produce that effect (and then comparing this to the treatment that is being investigated in the experiment).
Positive controls are often used to assess test validity. For example, to assess a new test's ability to detect a disease, then we can compare it against a different test that is already known to work. The well-established test is the positive control, since we already know that the answer to the question (whether the test works) is yes.