many, many, many,... many ways of going about achieving above...
screen capture software, would be first thought that comes to mind. what ever you see on your computer screen, it records it like a video, with optional audio as well. other words instead of a physical video camera setting beside you recording what you see on the screen. it is just a software you install with play,record, pause, stop buttons that show up on your screen, and no actual need for a video camera.
touch screen monitors for some might be easier, vs trying to draw with a mouse, some might use a "beh" tablet/pen combo *forget better term for them* instead of a mouse.
i would imagine there is software directly marketed to teachers. but if you can use "paint" that comes with every windows computer from 15 years ago to now... (mac's have like program), and screen capture that could work as well.
many web conferencing packages have some sort of annotation, / notation / markup. or some sort of drawing program in them as well.
computer screen resolution, and screen capture resolution, physical area you have to draw in / make notes in. can be dealt with. but ya need to know how to adjust settings both in microsoft windows, and what ever software you are using. along with what resolution you record at. can make a huge difference. and how things work and feel and getting stuff done.