8 bit computer graphics
Generally you will have a video RAM somewhere where the memory locations correspond to the pixels on the display.  This is generally special-purpose memory as it has to be read out continuously and written to the display.  In the case of a standard computer monitor, the video card will read pixel data out of memory and send it to the display via VGA, HDMI, or DVI.  In the case of a dot matrix display connected to a microcontroller, the video memory generally resides within the display control chip on the display.  The control chip continuously reads out the memory and drives the pixels on the display.  The CPU then just has to write to the video RAM where it wants the pixels turned on or off.  In computer monitors, several bytes correspond to each pixel.  In monochrome display, each bit in each byte could correspond to a pixel.  If you want to turn on some pixel x,y all you need to know is what memory location corresponds to that pixel and update it accordingly, either setting the color you want or turning on the specific bit that corresponds to that pixel.  To the microprocessor, everything is just a number; there is nothing inherent about the video ram that says 'this is video data' aside from the fact that it gets written to the display.  

Edit: as for your 'pong' game, this is a completely different animal.  The chip on there generates a composite video signal on the fly.  If you just want to draw some dots or lines, this isn't terribly complicated.  The basic idea is that a frame of video is represented by a series of lines, transmitted in sequence.  The lines are marked by horizontal sync pulses, and the frames are marked by vertical sync pulses.  In old CRT monitors, the sync signals are used to control the scanning of the electron beam across the display while everything in between the sync signals is sent directly to the electron gun to change the intensity.  All the microcontroller needs to do is generate the sync pulses at the right times and then set the output 'high' when the electron gun is currently pointed at the spot on the display that the controller wants to make light up white.  This can either be done with a video RAM, reading out one pixel at a time, or with a bunch of carefully-configured timers and counters.  This chip doesn't have enough RAM to store a whole frame, so most likely they are using the timer method.  The game has two paddles and a ball.  To generate the video signal, the controller generates one line of video at a time like so: generate an HSYNC pulse, wait until the first paddle column comes around, draw one point on the paddle if it currently intersects that line, wait until the ball column comes around, draw the ball if the ball is supposed to be on that line, wait for the second paddle column to come around, draw the paddle if it intersects the current line, wait the right amount of time for the next HSYNC pulse, and repeat.  There are some details with respect to generating the VSYNC signal as well as only drawing on the visible area of the display as well.