Dark under-eye circles begin in the capillaries, the tiny blood vessels that web the delicate skin around the eyes. Now, your capillaries are so small that red blood cells sometimes have to line up, single file, to get through. Frequently, red blood cells get lost and wander into the surrounding skin. This isn't a problem - it happens all the time - and your body has a mechanism to mop up these escapees. Enzymes in your body break down the red blood cells, including their hemoglobin (the molecule that gives them their distinctive red color).
No problem, right? Except for one thing: when hemoglobin is broken down, its remaining components have a dark blue-black color. Just like a bruise. So your dark under-eye circles are actually caused by leaky capillaries.