Partial pressure is a way of describing how much of a gas is present. All gases exert pressure on the walls of their container. This is because the molecules of gas constantly bounce off the walls. Partial pressure is used to describe a mixture of gases. It is defined as the pressure that any one gas would exert on the walls of the container if it were the only gas present.
This is defined more formally under Dalton’s law, which states that the sum of the partial pressures of all of the gases in a mixture will be equal to the total pressure of that mixture.
Confusingly, partial pressure is also used to describe dissolved gases, particularly in blood. In this case, the partial pressure of a gas dissolved in blood is the partial pressure that the gas would have, if the blood were allowed to equilibrate with a volume of gas. Why do we use this counter-intuitive system? The main reason is that when blood is exposed to fresh air in the lungs, it equilibrates completely so that the partial pressure of oxygen in the air spaces in the lungs is equal to the partial pressure of oxygen in the blood. This is demonstrated by the altitude oxygen calculator. You will notice that the partial pressure of oxygen in arterial blood is slightly less than the partial pressure of oxygen in the lungs: this is because there is always a little bit of blood that passes through the lungs without encountering an air space. This is called physiological shunt.