Blood is always red, actually. Veins look blue because light has to penetrate the skin to illuminate them, blue and red light (being of different wavelengths) penetrate with different degrees of success. What makes it back to your eye is the blue light.
Meanwhile, here's what happens to the blood in your body:
The heart pumps blood to your lungs to pick up oxygen. The oxygen-rich blood is then pumped out to your body through your arteries. It's bright red at this point. [How Much Blood Is in the Human Body?]
From your arteries, the blood flows through tiny blood vessels called capillaries, where it gives up its oxygen to the body's tissues. Your lips have a lot of these capillaries, which is why they're red.
Your blood, now exhausted of its oxygen, is dark red as it now returns to your heart through your veins.
This article has been corrected from the original version.