oxygen–hemoglobin dissociation curve
Picture a crowded bus that fills up in a peculiar way: the first passengers board slowly, then a rush fills most seats quickly, and the last few stragglers are hard to squeeze in. Hemoglobin loads oxygen with the same odd rhythm. The oxygen–hemoglobin dissociation curve is the graph of this behaviour — how full of oxygen hemoglobin gets (its saturation) at each level of oxygen pressure in the blood.
The curve is S-shaped, and that shape is the whole point. Where oxygen pressure is high (in the lungs), the curve is nearly flat near the top, so hemoglobin stays almost fully loaded even if lung oxygen drops a little — a safety margin for loading. Where oxygen pressure is low (in the tissues), the curve is steep, so hemoglobin readily gives up large amounts of oxygen for a small drop in pressure — efficient unloading exactly where oxygen is needed. The reason is cooperative binding: each oxygen that binds makes the next one bind more easily.
The curve is not fixed; it slides left or right with conditions. A right shift (caused by warmth, acidity, high carbon dioxide, or the metabolite 2,3-BPG) loosens hemoglobin's grip and releases oxygen more easily to busy tissues; a left shift tightens the grip, favouring loading. This responsiveness is why exercising muscle, which is warm and acidic, automatically gets more oxygen handed over to it.
The flat upper part also explains why oxygen saturation can stay reassuringly high until the underlying oxygen pressure has already fallen substantially.