Cross-sectional studies can be thought of as providing a "snapshot" of the frequency and characteristics of a disease in a population at a particular point in time.

This type of data can be used to assess the prevalence of acute or chronic conditions in a population.

Cross-sectional analysis studies the relationship between different variables at a point in time, For instance, the relationship between income, locality, and personal expenditure. Unlike time series, cross-sectional analysis relates to how variables affect each other at the same time. [Wikipedia]

Cross-sectional studies involve sampling subjects at random from a population and determining the levels of their explanatory and response variables. These are usually conducted retrospectively, based on large medical databases, at the health organization, state, or national level.

In these situations they have large numbers of individuals with extensive medical histories on each subject. Subjects are grouped, and associations between variables are investigated.

A Cohort Study is a study in which subjects who presently have a certain condition and/or receive a particular treatment are followed over time and compared with another group who are not affected by the condition under investigation.

For research purposes, a cohort is any group of individuals who are linked in some way or who have experienced the same significant life event within a given period.

There are many kinds of cohorts, including birth (for example, all those who born between 1970 and 1975) disease, education, employment, family formation, etc.
Any study in which there are measures of some characteristic of one or more cohorts at two or more points in time is cohort analysis. [Social Research Methods]

Cohort studies are generally prospective; it means such studies involve identifying subjects based on the level of their explanatory variable, and obtaining the corresponding response outcome.
But they could be retrospective too.

These studies usually involve following the subjects over a period of time to determine their outcome.

For instance, many studies have been conducted to compare the rates of breast cancer in women with breast implants and women without breast implants.
Women were identified as either having breast implants or not (explanatory variable), and were followed over time to see whether or not they were diagnosed with breast cancer (response).

Cohort studies are common when it is unethical to assign a condition (such as smoking or breast implants) to subjects, but it is possible to identify existing populations of such subjects.

Cohort studies have to have enormous sample sizes when the outcome of interest is rare in the population.

Case-control is a type of epidemiological study design.
Case-control studies are used to identify factors that may contribute to a medical condition; by comparing subjects who have that condition (the 'cases') with patients who do not have the condition but are otherwise similar (the 'controls'). [Wikipedia]
Then all subjects are asked about their status considering some risk factor of interest.

Case–control studies are generally retrospective.

Case–control studies are commonly used when the response of interest is very rare in the population of interest.

Suppose, for instance, a study involves children born with defects (cases) and those without defects (controls).
Mothers of the children with defects may be more likely to recall use of a prescription drug, since they would probably have spent more time contemplating their pregnancy than the control mothers.

Case-control studies use patients who already have a disease or other condition and look back to see if there are characteristics of these patients that differ from those who don’t have the disease. [Wikipedia]



Generally speaking, case/control studies are the weakest at determining a causal relationship, but may be the quickest and cheapest way to determine risk factors that may be then studied prospectively.



Some references explain the weaknesses and strengths of case-control studies like this:

• Strengths of Case-Control Studies:

With enough subjects in the study and careful selection of controls, case-control studies provide a cost-effective way to study cancer.

• Weaknesses of Case-Control Studies:

Quick: list the kinds of foods you ate most often ten years ago. Like eyewitness testimony in a courtroom, case-control studies depend on our unreliable memories. In case-control studies, cases and controls may remember their past diets differently.

Some new scientific developments, such as biomarkers of dietary intake act like fingerprints of the foods we eat regularly, can help to avoid this problem