These are examples of FreeSurfer Group Descriptor (FSGD) files and contrasts for testing various hypotheses. FreeSurfer can automatically construct design matrices from an FSGD file. The FSGD file is usually easier to construct than the design matrix. The FSGD file is used as input to several FreeSurfer programs (mainly mri_glmfit and mris_preproc). Note that you can always specify you own matrix using the --X option to mri_glmfit.
These examples are not meant to be exhaustive, nor are they meant to teach you statistics. See these slides for more of the statistics background on group analysis and how it applies to FreeSurfer.
An alternative to constructing your own FSGD files is to use QDEC, though QDEC cannot be used with more than four groups and/or more than two continuous variables. In the examples below, the term "QDEC-able" appears after those examples that can be analyzed in QDEC.
Below, DODS refers to "Different Offset, Different Slope" and DOSS refers to "Different Offset, Same Slope". Refer to the DODS vs DOSS page for more information.