correlational
ethnographic
longitudinal
case studies
cross sectional
experimental
mixed methods
survey
I'll list what each of these are and you can think about the pros and cons.
Ethnographic - this is commonly used in sociology, in which the researcher immerses him/herself in the culture or subgroup being studied. This would be akin to a sociologist studying the culture of Harlem going and living in Harlem for a year and taking notes of his/her observations. This is usually observational and not experimental, as it's unethical to manipulate humans without their knowledge.
Longitudinal - this is commonly used in medicine, where the researchers track the subjects' outcome over an extended period of time - usually years. So for example, one could study the effect of trauma during birth on the development of mental health issues. In a longitudinal study, the researchers would take a sample of infants who endured trauma during birth along with a control group and have them come in every year/six months for a set number of years to assess whether they develop mental health issues. This is a powerful technique because it eliminates the critique that an effect could take time to manifest or that the effect had gone away by the time a study was performed on the subjects in which the study measured only a single time point.
Case studies - these are commonly used in medicine, especially psychiatry. In these types of studies, the researcher or physician focuses on one patient/subject. The subject's background, history, and symptoms/signs are extensively documented. Oliver Sacks does a lot of this - his books are extremely interesting to read, even for a lay reader and I highly recommend them.
Cross-sectional - this is the type of study whose criticism can be addressed by a longitudinal study. That is, it measures effects at one time point over a large sample/cohort. So using the example above of trauma during birth and mental health issues, a cross-sectional study would take a sample of infants who had suffered trauma during birth along with a control group and assess their mental health at 25 years of age. It's a "cross-section" of the life course and therefore is a "cross-sectional" study.
Experimental - these are the kinds of studies most people think about when they think about science. It's what's done in the laboratory. That is, you manipulate one variable, e.g. diet, and measure the effect, e.g. mouse weight. These are carefully controlled and are generally considered to be the gold standard for testing cause and effect.
Surveys - I don't think you need an explanation for what this is.
Mixed methods - exactly as it sounds - mixed methods employ a mix of two or more of the methods listed above. So for example, one could perform an ethnographic case study in which the researcher integrates into the subject population but focuses on a single subject during that time.
