Why does air pressure decrease with altitude?

The air pressure at a given point is the weight of the column of air directly above that point, as explained here. As altitude increases, this column becomes smaller, so it has less weight. Thus, points at higher altitude have lower pressure.

While gravitational force does decrease with altitude, for everyday purposes (staying near the surface of the Earth), the difference is not very large. Likewise, the centrifugal force also does not have significant impact.

I edited this question on the first day, in response to a few comments that pointed out a misunderstanding, but it didn't register. I sincerely apologize for that.

As pointed out by other answers, the pressure due to any fluid, compressible or not, increases with depth. This is due to the greater mass and thus weight of the fluid above.

What's interesting is that the pressure of water increases linearly with depth, while that of air does not.

The gravitational field strength drops down to only 88% even at the height of the ISS. The drop in pressure has more to do with the fact that unlike water, air is a compressible fluid. As you move further down the atmosphere, there is a greater weight of air above pushing down on the air below, so the density, and thus the air pressure, increases. Basically, the density $\rho$ is a function of $h$. so you have to integrate density over the height instead of simply multiplying.


or $$P=\int g\rho\mathrm{d}h$$ if you want to account for the change in gravitational field, however small

As you go higher, there are less air molecules (less weight) on a given area this is basically one reason why it decreases.

From the barometric formula, one can get the relation between the pressure and altitude. It's defined as

$$P = P_{0}e^{-\frac{mgh}{kT}}$$

so the relation between pressure and altitude is $P\propto e^{-h}$. Thus, as we go to higher altitudes pressure will exponentially decrease.