Basic handles questions

Another example, if you are drawing with the pen tool,  say  to make the top  or bottom of a heart shape, you'll need the path to change direction and to do so you break the handles  to form what's called a cusp point, by holding down the Alt (option) key, whilst the mouse is still held down.

Like in this youtube video

at 3.10

Dansky is playing the Bezier game - link in his video  Description - to show the idea.

MASTER the Pen Tool with the Bézier Game - YouTube

You might want to move them independently if you want both sides from the anchor point to curve up from it rather than down, for example. Or perhaps not be equally curved on each side of the anchor point. To do this you can make a corner point, as Monika says. You can use the Shortcut of using the Alt/Opt key while drawing with the Pen tool to independently adjust the handles. Or if you have already drawn something, you can use the direct select tool and use the same shortcut to make move them independently.

To get them to move independently, you need to make a corner point. Use the anchor point tool to break the handles.