There are two kinds of anchor points:
Corner points (with no handles) and Smooth points (with handles).
The handles control the curves and since corner points have no curves they have no handles. You can convert one kind of anchor point to the other in a number of ways:
- Use the Convert Anchor Point tool (SHIFT+C).
- With the Pen tool (P) active, Alt+Click the anchor point.
- Select the anchor point with the Direct Selection tool (A) and use the Convert buttons in the Options Bar
The little "ball" you see on corner points is the Live Corners widget, which allows you to modify the corner and as the name suggests, only works on corner points.