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Human blood is red because of the iron in the haemoglobin. This gives red blood cells their red colour. Veinous blood is much darker than arterial blood, and your teacher is right that oxygenated (arterial) blood is 'redder' than deoxygenated (veinous) blood. However, human blood is not blue. Some animals do have blue blood though.
Blood is a specialized fluid in your body that has four main components, each with a different function: plasma, red blood cells, white blood cells, and platelets. Watch this video to learn more about how these components work.