Based on what I learned from the textbook, there are several different study designs, such as case-control study, case-cohort study, case-crossover design, and cross-sectional study.  let me briefly introduce them one by one.
Firstly, what is a case-control study? Basically, we need two groups of people, one group of people with the disease (called cases), and the other don't have the disease (called controls). Then we compare these two groups and try to figure out the possible relation between an exposure and a certain disease. The major limitation of this design is recall bias between people who have the disease and try to connect the disease to the exposure intensionally.
Secondly, what is a case-cohort study? In this type of study, we basically select a certain population and follow them over certain time period. This type of study is a good fit for finding risk factors for a certain disease without recall bias, since people were followed from baseline.
Thirdly, what is a case-crossover design? Normally, this is used for studying the acute outcomes with short exposure. Each people in this study represents his/her own control. Recall bias is still the major problem for this type of study.
Fourthly, what is cross-sectional study? The main feature for this type of study is both exposure and disease are examined simultaneously for each people at one time point. This is also called prevalence study, since the determined cases of disease represents the prevalence of this disease at one time point. The major problem for this type of study is missing temporal information. This means we don't have any time series information to know whether exposure happens before the disease or the other way around. So, we could only address the association between exposure and disease, but not the causality.
1. Gordis L. Epidemiology. 4th ed. Elsevier. 2008. ISBN: 9781416040026