You are creating mask paths using the Rotobezier mode (see the checkbox in the option bar, to the right of the tools).
Masks (and shape) paths that you generate with pen tools can operate in two modes: Bezier and Rotobezier. Since Rotobezier was introduced, After Effects defaults to that mode.
When you create and edit masks in Bezier mode, the curve tension is defined by bezier handles (unless it's a corner vertex, of course).
When you create and edit masks in Rotobezier mode, the curve tension is defined by a tension setting in each vertex, which you can tweak with the Convert vertex tool or by control+alt+dragging (Win) / command+option+dragging (Mac). Rotobezier vertices have a default tension setting which produces averaged curves from vertices automatically.
As a general rule, Bezier is more precise spatially but less predicatable in terms of interpolating over time, compared to Rotobezier.
To select a specific vertex, just use the selection tool.
For more information, see the AE Help section on Editing and animating shape paths and masks
You use a combination of a (arrow tool) and g (pen tool) to adjust your mask splines. Also use shift and control for selecting knots on the mask spline. When you select a knot (think it might have to be a single knot selection), it should show the bezier handles.
When you select a knot (think it might have to be a single knot selection), it should show the bezier handles.
Not if the mode is set to Rotobezier, which uses a different method from Bezier handles to define tension.
Thanks everyone for the help.
I think I've got a handle on things now...