This is a list of each and every type present in the game. Each page includes every puppet that holds that certain typing, as well as some other details.
For the relation between each type, see the Type Chart.
For how one specific puppet fares against each type, see that specific puppet's page, or use this application. (The application can also be used for team analysis.)
However, please note that the listed application does not account for abilities, and that, right now, not every puppet's page is complete. The latter should be fixed soon, however.