The simple answer is bitmaps, in fact this would also apply to fonts on terminals in the early days.
The original GUI's, things like Xerox Parc's Alto GUI were based on bitmap displays, and the graphics were drawn with simple bitmap drawing tools and graphics libraries, using simple geometry to determine shapes like circles, squares, rectangles etc, and then map them to display pixels.
Today's GUI are the same, except with additional software and hardware that have sped up and improved the process, and performance of these GUIs.
The fundamental mapping of bits e.g. 10101010 to pixels is dependent on the display hardware, but at a simplistic level, you would provide a display buffer in memory and simply populate it's bytes with the display data.
So for a basic monochrome bitmap, you'd draw it by providing bits that represented the shape you want to draw, you can either position these bits, like this, a simple 8x8pix button.
01111110
10000001
10000001
10111101
10111101
10000001
10000001
01111110
Which you can see easier if I render it with # and SPACE instead of 1 and 0.
######
# #
# #
# #### #
# #### #
# #
# #
######
Which as a bitmap image would look like this : http://i.imgur.com/i7lVQ.png (I know it's a bit small :) but this is the sort of scale we would've begun at, when GUI's were first developed.)
If you had a more complex (e.g. 24 bit color display, you'd provide each pixel using a 24bit number.)
Obviously some bitmaps cannot be drawn manually (for example the border of a window), like we've done above, this is where geometry comes in handy, and we can use simple functions to determine the pixel values required to draw a rectangle, or any other simple shape, and then build from there.
Once you are able to draw graphics in this way on a display, you then hook a drawing loop onto a system interrupt to keep the display up to date (you redraw the display very often, depending on your system performance.) This way you can make it handle interaction from user devices, e.g. a mouse.
Back in the early days, even before Xerox Parc / Alto there were a number of early computer systems which had Vector based displays, these would make up an image by drawing lines on a CRT representation of a cartesian plane. However, these displays never saw mainstream use, except perhaps in some early video games, like Asteroids and Tempest.