Answer:
First look up the ideal gas law: PV=NRT, where P=pressure, V=volume, N=amount of gas in moles, R=universal gas constant, and T=Temperature in degrees K. See Ideal gas law - Wikipedia for details.
Then consider the following:
Inside a tire, V and R are constants, while P, N, and T vary.
What makes a tire seem hard or soft is the difference in pressure between the air in the tire and the air around the tire.
A so-called flat tire is not devoid of air; it just has the same pressure inside as outside.
When you pump up a tire, you are increasing N by forcing more air molecules into the tire. As V is a constant volume, you create a pressure difference between inside and outside.
This pressure difference varies with how much you pump the tire (P), as well as the temperature around the tire (T) and the air pressure around the tire (atmospheric or “ambient” pressure).
The fact that an inflated tire is in balance with ambient pressure — with the added influence of PV=NRT — has some counter-intuitive results. For example,
I’ve had bike tires blow out by being left in a hot car. Why? The pressure in the car is the same, but the pressure in the tire increases as T increases until it’s beyond the safe limit.
More rarely, I’ve heard reports of bike tires blowing out by being carried in the unpressurized hold of an airplane. In this case, the tire doesn’t get hot; instead, the pressure outside the tire decreases, until the tire can’t hold in the pressure. In actuality, this is really hard to do; the tire has to be over-pressurized to start with; the difference in ambient pressure is not enough to blow out a typical bike tire.