I have two layer, one (A) has point features and the other (B) has polygons. I need to produce a new table with some (the ID) or all the fields from the A table and some (the area code) or all from the B table where the B polygon contains the A point.
It's possible to do a join on location ("where the B polygon contains the A point") using "Join attributes by location" from "Vector" menu. The resulting layer will contain all attributes from both layers.
A faster alternative is SAGA Add polygon attributes to points which is available through the Processing toolbox in QGIS.