What does a triple integral represent?

From my understanding if the integrand is 1, then it gives you the volume of the region defined by the bounds. But what does the value of a triple integral represent if the integrand is a function for a surface in space?

You can think of the integrand as the "density" of the region and the value of the integral as the "mass" of the object.

For example, $$ \int_0^1\int_0^1\int_0^1 1 \, \text{d}x \, \text{d}y \,\text{d}z $$ can represent the volume of the unit cube within the region $0\le x\le 1$, $0\le y\le 1$ and $0\le z\le 1$.

For $$ \int_0^1\int_0^1\int_0^1 (x^2+y^2+z^2) \, \text{d}x \, \text{d}y \, \text{d}z\ , $$ you can think about it as the mass of the same cube where its density is given by the function $f(x,y,z)=x^2+y^2+z^2$. This means that the cube is light near the origin and is getting heavier as you move away from it.

In general, $f$ can be negative so you must consider the signed-mass which means that the mass can be negative somewhere...

When $1$ is your integrand, you have these geometric interpretations:

$\quad\int dx$ is to length

$\quad\iint dA$ is to area

$\quad\iiint dV$ is to volume

When your integrand is some function, then you've likely heard:

$\quad\int f(x)\ dx$ is the signed area underneath the curve.

But wait, you say, why is it that $\int dx$ is a length, but $\int f(x)dx$ is an area? First, it's all an interpretation. One could say that $\int dx$ is the area under the curve $f(x)=1$. The difference is that if $f(x)\ne1$, then we're assigning some non-trivial value to every point in the evaluated space. Note that if $f(x)$ and $x$ have physical units $[f(x)]$ and $[x]$ respectively, then that "area" has units $[f(x)]\times[x]$. For example, integrating a velocity function over some period of time $\int_{t_0}^{t_f} v(t)\ dt$ would give you a change in displacement.

So this is a sort of density. At every point in the interval, you're summing how much of the integrand is present. You're finding how much "stuff" is associated with the interval of integration. With that in mind, let's revise our previous statement:

$\quad\int f(x)\ dx$ is the signed amount of stuff associated with the interval of integration.

It follows that

$\quad\iint f(x,y)\ dxdy$ is the signed amount of stuff associated with the region of integration.

This could certainly be a volume, but we shouldn't limit ourselves. Integrating a surface charge density over a region would give you the total amount of charge, wouldn't it? That's not a volume. What we're really doing is assigning a weighting factor to every point in the region and summing all of these contributions via integration.

$\quad\iiint f(x,y,z)\ dxdydz$ is the signed amount of stuff associated with the volume of integration.

If your integrand was a mass density $\rho(x,y,z)$, then integrating over the volume would give the total mass.

In terms of understanding what an integral actually does, what I presented is mostly a heuristic, but I hope it's a useful one. It certainly rings true in physics. If there's one thing to note, it's that if you're integrating quantities with physical dimensions, then the units of your result will have units equal to that of the integrand times those of the differential elements.

That is, if $I = \int f d\tau$, then $[I] = [f][d\tau]$.