A research study comparing the risk of developing lung cancer, between smokers and non-smokers, would be a good example of an observational study.
The main reason for performing any observational research is due to ethical concerns.
With the smoking example, a scientist cannot give cigarettes to non-smokers for 20 years and compare them with a control group. This also brings up the other good reason for such studies, in that few researchers can study the long-term effects of certain variables, especially when it runs into decades.
For this study of long-term and subtle effects, they have to use pre-existing conditions and medical records. The researcher may want to study an extremely small sample group, so it is easier to start with known cases and works backwards. The thalidomide cases, for example, are an example of an observational study where researchers had to work backwards, and establish that the drug was the cause of disabilities.
For example, in the smoking example, if the researcher found that there is a correlation between smoking and increased rates of lung cancer, without knowing the full and complete background of the subjects, there is no way of determining whether other factors were involved, such as diet, occupation or genetics.
Randomization is assumed to even out external causal effects, but this is impossible in an observational study.
There is no independent variable, so it is dangerous to assume cause and effect relationships, a process often misunderstood by the mass media lauding the next wonder food, or sensationalizing a political debate with unfounded results and pseudo-science.
Despite the limitations, an observational study allows a useful insight into a phenomenon, and sidesteps the ethical and practical difficulties of setting up a large and cumbersome medical research project.