What are all the software and hardware needed to produce a video of this kind?
This looks like she used a graphics drawing pen tablet (e.g. a Wacom Pen Tablet) and recorded her screen with a screencapture software such as Fraps. To have the person appear in the lower right like this, you will need a well-lit greenscreen to key out the background. To produce the video from your recordings, you will need NLE software. Since there aren't any special effects in this video, you don't need a professional program; the only requirement would be the greenscreen processing capability.
Alternatively, you could also create the screen writing effects completely digitally with an animation/SFX program like After Effects. However, this is pretty difficult and it will certainly take some time to learn the required skills to produce a video like this.