In an experimental study, the variables are the independent and dependent variables and as little other confounding factors as possible. This design frequently uses experimental and control groups, although there are sometimes variations in how the measurements are taken within these groups. Subjects are randomly assigned to the different groups and the experimental group or groups are given the treatment (independent variable) and the control group -- usually only one is needed -- is not given the treatment (independent variable). For all groups, the same dependent variable or variables are measured. The purpose of this variable set up is to suggest that the independent variable causes a change in the dependent variable.
Quasi-experimental designs use the same variable setup as an experimental design (the independent and dependent variables) but do not use random assignment of subjects. These experimental designs are important when the subjects cannot be randomized, such as using an intact group like a classroom. Quasi-experiments can still use control groups, although subjects are not randomly assigned to this group. There are different sub-types of quasi-experiments that vary how the subjects are measured during the experiment. For example, the dependent variable may be measured on a pre-test given before the treatment, then again on a post-test given after the treatment. You can then suggest that the change (if any) in the dependent variable is a result of the treatment.
Factorial designs add a third variable to the experimental design, sometimes called a "moderator" variable. The independent and dependent variables remain the same and are measured in the same way, but a third variable is added to examine the combination of this new variable on the dependent variable. In this situation, you are likely to end up with more than two groups. For example, if you wanted to measure the effects of a new drug but you think that blood pressure might affect how the drug works, you can create a control group (no drug), an experimental group with participants with high blood pressure who get the drug, an experimental group with participants with average blood pressure that also get the drug, and an experimental group with low blood pressure that get the drug. This will allow you to see if the drug has an effect and if blood pressure influences the effectiveness of the drug, which is referred to as an interaction effect.
Sometimes there is only one participant or one group of participants that are of interest to the researcher. Such a situation may be very rare occurrences such as participants with rare diseases or disorders, participants who experienced a special situation, or in psychological studies examining behavior treatments. These studies are called single-case or time-series experiments and should not be confused with the "case-study" methodology in qualitative research. This design typically uses the independent and dependent variable set up of an experiment, and uses the subject as the control by systematically removing the treatment. A good example of this is the ABAB study. A pre-test is taken to establish a baseline measurement before the treatment is applied (A). Then the treatment is administered (B) and measurements are taken again. Next the treatment is removed (A) to see if the measurement returns to the baseline level. Finally, the treatment is administered again (B) to see if the change in the dependent variable can be seen again.