Select all polygons that are completely within a circular buffer

You should use "Vector->Research tools->Select by Location" and check the "Use selected features only" box. You will want to include ONLY:

Include input features completely within the selection features

https://docs.qgis.org/2.6/en/docs/user_manual/processing_algs/qgis/vector_selection_tools/selectbylocation.html

Just to add to the already given great answers, you could also use the Select by location tool from the Processing Toolbox to give you more options on how you want to select your objects:

There is a spatial query plugin that makes this possible. It has a 'within' operation. You can see the QGIS docs on the plugin here.

Edit: to add that you'll have to add the plugin via the Plugin Manager. If you are unfamiliar with this in QGIS, here are the basic docs.