How does the HD44780 LCD work at the low level?
The HD44780 is a character LCD controller, which means you write ASCII codes to the LCD for the text you want to display. If you want to control pixels individually you have to go for a graphic LCD.

The HD44780 does offer the possibility to display some limited text-based graphics, though; you can define 8 custom characters, but at 5 x 7 pixels per graphic that won't allow you to display anything like an picture.  

The HD66100F is an 80-segment LCD driver for e.g. a 10 digit 7-segments display.