Before talking about this technique let’s clarify that when the name repeated measures MANOVA is used it commonly refers to a repeated measures ANOVA in the sense that there is only a single dependent variable or outcome of interest. For example, a researcher who wants to compare different treatments for diabetes has taken 4 weekly measurements of blood sugar to a series of patients.
The fact that the term MANOVA (i.e. Multivariate Analysis of Variance) is used to describe this situation is because one can see it as having multiple dependent variables, one per each time point. To continue with our example, our researcher would have one variable for blood sugar taken at time 0, another for blood sugar at time 1 and so on up to 4 dependent variables. In fact, this type of database structure is known as wide as opposed to long shape, where each repeated measure occupies a separate row and there is only one column for the dependent variable.
Some statistical packages require data to be entered in a particular shape. For example, SPSS requires wide format to conduct repeated measures ANOVA. Other packages such as Stata allow you to use the anova command and declare repeated measures as part of the command options. This command requires your data structure to be long. Or if you have data in wide shape Stata allows for the use of the manova command in exactly the same fashion explained above for SPSS.
Going back to when a repeated measures ANOVA/MANOVA is used, it is as follows: you are interested in assessing the impact or effect of different factors or combinations of factors in your study outcome. The difference from a traditional ANOVA is that you have taken more than one measurement of your outcome for each of your study subjects. Remember that factors are defined as categorical variables, for instance drug treatment and that your outcome variable needs to be continuos. You can find ANOVAs run on a Likert scale which can be considered an approximation to a continuos variable (although there is some controversy around this topic).
Finally, an important thing to take into account when working with repeated measures is defining the structure for the covariance matrix. The simplest structure is the Identity which assumes that repeated measures of a same subject are independent and with constant variance across time. This is a not very realistic assumption since you would expect that measures taken from the same subject should have some degree of dependency. The second simplest structure is called Compound Symmetry or variance components, which requires the estimation of two parameters. This structure assumes constant variance and covariance. The latter means that the correlation between two observations of a same subject is equal for all pairs, it does not depend on the lag between them. This is not a very realistic assumption either. A more realistic structure is the autoregressive order 1 (AR(1)) which also assumes homogeneous variance across time but where the covariance between the observations on the same subject decrease as the lag between them increases. Finally, the unstructured covariance structure specifies different variance and covariances. It is the most general of the structures but demands the estimation of a large number of parameters.