https://www.allegro.cc/manual/5/al_draw_line
thickness - Thickness of the line, pass <= 0 to draw hairline lines
It's a line that recedes with time 
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It basically means that it's thickness is infinitesimal. You still see it, but e.g. if you zoom in with a transformation, it won't get thicker. At the same time, it won't disappear if you zoom out either. It's in some sense the smallest thing the device can draw to represent the mathematical line segment.
As a side effect, hairlines are undefined to as in which pixels are lit up. If you specify a thickness, the GPU guarantees that each pixel whose center lies inside of the line is lit. If you draw a hairline, the GPU is pretty much free to do whatever it wants.
As a side effect, hairlines are undefined to as in which pixels are lit up.
That's mostly because OpenGL neglected to specify this, they could have easily came up with a definition (like the one A4 uses, which is just nearest pixel to the line).
It's a line that recedes with time .
