When the hemoglobin in the red blood cells binds with Oxygen, the Oxygen borrows an electron from the Iron atom at the core of the Heme. This changes the molecular orbitals of the electrons in the hemoglobin. The hemoglobin is then less interested in red photons of light (it doesn’t absorb them as much), and we perceive this as it changing colour to a brighter shade of red.
But in case you don’t know what hemoglobin is – it is a protein found in red blood cells which contains an iron ion. This iron ion binds to and carries oxygen from the lungs to the rest of the body where it releases the oxygen to burn nutrients to provide energy to power the functions of the cells.
Because 35% of a red blood cell is hemoglobin and because blood is made up of of mostly red blood cells this makes the overall colour red.
Its this protein which makes blood red – in the way the Jarvist described!
The way that you perceive (or see) colours though is because light from the sun or from light bulbs looks white-ish when it leaves from there but it is actually a mixture of colours, exactly like a rainbow. When it lands on an object, for example some blood, most of that light is actually absorbed by the object but some isn’t. In the case of blood, as Jarvist said, the red part of light is not absorbed and is reflected – it is this part of the light that reaches our eyes. And, because our eyes are designed to detect different colours, those sensors in the eyes send a message to the brain that the object is red.
Did you know that mice cannot see red – their eyes just don’t have the same cells we do to detect it; so, if you put them in a clear red plastic box they think that they are in the dark!