A Retrospective Study is a kind of research design where the investigators study a phenomenon by looking back at events that have already happened.
For example, to look at the relationship between serial killers and child abuse, the investigators will identify convicted serial killers, and find out which of them have a past history of being abused as children.
A major advantage of a Retrospective Study is that it allows the study of rare phenomena without having to use a very large sample size, and without having to spend a long time following the subjects to find out the outcome. If you were to do a Prospective Study on serial killers and child abuse, you would have to recruit an extrermely large sample of abused children and follow them for several years in order to find out which ones turn out to become serial killers. That would be very expensive and time-consuming.
A disadvantage of this kind of study is that it is prone to bias. In this example, it is possible for the researchers to tend to select participants who already have a known history of being abused. Also, the accuracy of the gathered data is only as good as the accuracy of available records.