Multivariate designs have more than one dependent variable. Multiple regression and multiple correlation, involving one dependent variable and several independent variables, however, are also considered multivariate statistics.

Multiple Regression: goal is to explain as much of the variance in the criterion variable (Y - the dependent variable) based on a set of predictor variables (Xs - predictor variables).

Discriminant Analysis: basically Multiple regression, with a categorical dependent variable.

Canonical Correlation: looks at the relationship between a set of predictor variables and a set of dependent variables by creating one new predictor variable and one new dependent variable and related these canonical variates. [These new variables are the predicted scores based on multiple regression.]

Multivariate Analysis of Variance (MANOVA). Used when you have more than one independent variable and more than one dependent variable that you believe are related (i.e., not independent). This is preferred over multiple ANOVAs for the same reasons as ANOVA is preferred over multiple t-tests. As with multiple t-tests, multiple ANOVAs could lead to probability pyramiding (inflating alpha error rates). Furthermore, it may be the combination of dependent variable scores that is influences by the independent variables and separate ANOVAs would miss this (just as separate t-tests wouldn't detect interactions among IVs). (This can be used to analyze within subject designs where each repeated measure is treated as a separate dependent variable. SPSS does this.)

Log-linear analysis. This non-parametric statistic is basically a multivariate Chi-squared. Very useful when dealing with several categorical variable.

Path Analysis. This technique uses multiple regression methods to examine hypothesized causal relationships among variables with only correlational data. In essence you can see how well your theoretically derived model accurately describes or accounts for the relationships among variables. This can also be compared to competing theories about the relationships among variables. In some cases time of measurement can be used to limit directionality problems.

Factor analysis is a multivariate form of data reduction. Factor analysis is typically use to extract a relatively small number of underlying dimensions or factors that can account for relationships among measures (see example from text)

These are all very powerful and easy statistics to use, and misuse. To use these the techniques appropriately depends upon careful research design and thought.