In a univariate study, you examine the effects of the independent variable on a single dependent variable. For an experimental study, the experimental group is given the treatment (a new drug, for instance) and the control group is not given the treatment. You measure the same dependent variable for each subject in each group (blood sugar, heart rate, grades or attitudes, for example). With univariate statistics, you try to establish a causal relationship between the independent variable and a change in the dependent variable. Did the drug work? In univariate studies, you can also have more than one independent variable (a drug cocktail, for example), so long as there is still a single dependent variable.
Bivariate studies measure the relationship between two variables. Neither of the variables being studied is an independent variable, so the procedure is not experimental, such as in univariate studies. Correlations are common bivariate tools and are used to study how one variable influences the other. For example, if you wanted to see how family income influenced graduation rates, you could use a bivariate correlation to examine the two variables.
Multivariate studies are similar to univariate studies, but they have more than one dependent variable. For example, if you wanted to examine the ability of three new chemicals to clean an oil spill, the three chemicals would be your independent variables. You could measure the chemicals' dispersant properties, detoxification of the oil, toxicity of the chemical and effect on the environment as your dependent variables. You would then use a multivariate statistical analysis to examine the relationships between all of the variables.
In the classic univariate study, a group of randomly selected subjects is assigned to a control or treatment group and examined on a single factor (dependent variable). In subjects like psychology, you are typically interested in more than one factor and want to try several different treatment methods. Say you wanted to study the efficacy of a new behavioral treatment on people with depression. The people gathered for the study will likely come with a host of differing qualities, all of which could be classified as an independent variable. Additionally, the new treatment may affect other aspects of people as well as depression, such as self-esteem or self-image. This multivariate study is much more realistic than assigning people to groups and hoping they all turn out as you predicted.