About corners versus curves
Each anchor point (or node) on a Bezier path always has two control handles: an "in" handle that controls the tangent of the curve leading into the point, and an "out" handle that controls the tangent of the curve that leaves the point. Which direction is "in" and which is "out" is usually irrelevant when you modify a path, but the difference relates to the path's direction.
There are three types of anchor points:
Smooth point Each control handle of a smooth anchor point is extended and the two handles are pointing in exactly the opposite direction (180 degrees) from each other. When you first draw a curve with the Pen tool, its handles are symmetrical. In other words, they're positioned the same distance away from the anchor point.
Symmetrical handles on a Smooth point After dragging one handle with the Direct Selection tool, the handles are no longer symmetrical, but the point is still smooth
Unconstrained (or Cusp) point An unconstrained or cusp point has at least one extended handle, and each handle can be any distance or angle from its anchor point.
Cusp point with two extended handles (top) versus only one extended handle (bottom)
Corner point In a corner point, the two handles are fully retracted. Therefore, they're sitting "inside" the anchor point itself and cannot be selected.
For more information about how to draw these curves, see Use the Pen tool.
Converting anchor point types
You can change the type of an anchor point by going to the Toolbox, and selecting the Convert Anchor Point tool (or pressing SHIFT+C).
When you click a B-spline curve node with the Convert Anchor Point tool, it changes to a corner point.
When you click a corner point and the two adjacent nodes are also corner points, the node turns to a B-spline curve node.
When you click a Bezier smooth or symmetrical point, the control handles are fully retracted, and it becomes a corner point.
When you drag any kind of Bezier node, it extends both handles and becomes a symmetrical point.
You can drag a control handle of a symmetrical or smooth point to release the anchor point and then move the handles independently.
You can convert a smooth point to a cusp point by using the Direct Selection tool, holding down the ALT key, and then dragging one of the tangent handles.