Why is the lithostatic pressure gradient 1 psi/ft? How is it derived from density?
Lithostatic pressure might be roughly 23 kPa/m (about 1 psi/ft) in young basins dominated by siliciclastic rocks, but it varies substantially. Pressure has a linear relationship to density. Let's explore the relationship, and the variance.

First some definitions:


Lithostatic pressure — vertical pressure due to the weight of rock and fluid. 
Hydrostatic pressure — vertical pressure due to the weight of fluid only.
Fracture pressure — the pressure at which the rock fractures.
Pore pressure — the pressure in the pores of a rock; usually close to hydrostatic.


There's a lot of jargon in this field — lithostatic pressure is sometimes called confining pressure, overburden pressure, vertical stress, total stress, and various other things. I hadn't heard of geostatic pressure before, and a quick bit of research revealed that some people call the pressure due to rock and fluid 'lithostatic' and some call it 'geostatic', reserving 'lithostatic' for the pressure due to the dry rock only. In my experience, it doesn't pay to be pedantic about these inconsistencies, all you can do is spell out what you mean. 

We'll do a quick derivation (there's a more comprehensive one here). Pressure $P$ is force $F$ per unit area, and Newton told us that $F = ma$, where $m$ is mass and $a$ is acceleration. Since we're interested in vertical pressure, acceleration is due to gravity so we can use $g$. We know the mass of a rock normalized to volume as density $\rho$, and we're interested in a depth $d$, so:

$$P = \rho d g$$

The units are $[\mathrm{kg/m}^3] \times [\mathrm{m}] \times [\mathrm{m/s}^2]$ or newtons per square metre, N/m$^2$, which we call pascals, Pa, for convenience (10 kPa is 1.45 psi). Note that in the real world density varies with depth, so you'll need to integrate.

Because the pressure increases roughly linearly with depth, it's often convenient to talk about the pressure gradient, which is usually expressed in units of kPa/m, and of course is just the gradient of a line on a pressure–depth plot, like in the example below from a fictional offshore well.

So where does 23 kPa/m come from? If we take an approximate average density of sedimentary rock as 2300 kg/m$^3$, then $P = 2300 \times g$ Pa/m, or about 23 kPa (we often round $g$ to 10). Different mineralogies, different porosities, different fluids — all these will change the lithostatic pressure gradient.