You can use Gmm when you want to have the lagged dependent variable as your independent variable or when the dependent variable has high persistence or long memory. If that is the case then, fixed and random effects estimate would be inconsistent. Better to use 3 step GMM estimation procedure.
You should check out the articles of Arellano & Bond (1991) and Arellano & Bover (1995) then check out GMM module of Eviews. I hope they will help you !
As for cross section, there is not just one GMM for panel data. You can actually interpret the RE and the FE as GMM. Furthermore there are, for instance, the Difference and System GMM which take into account other specificities of the data. So you would probably need to know, what properties your data has in order to identify the appropriate method. Otherwise you can maybe specify your question for a more detailed answer.
Generalised Method of Moments (GMM) is the estimation technique to evaluate the relationship between dependent and independent variables in a specified model