Moments in math, they describe the "shape"? There's this one tantalizing line from the Wikipedia article:

In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points

Can someone expand on that?
Otherwise moments in math don't seem to arise from anything related to the figures of which you find the moments of, but rather are just "nice things to know" about a figure.
E.g. the first moment:
$$
\iint_R y \rho(x,y) dA
$$
(That's supposed to be over a region R)
Gets you the first moment with respect to $y$.  Which has no meaning in and of itself, but happens to be:
(Physics)


*

*torque (what it takes to rotate R pushing from the y side)


(Statistics)


*

*the mean (where $\rho$ is a probability function over R)


The second moment gets you the variance (Statistics) and the Moment of Inertia (Physics), etc.
So it looks like this form just "shows up" in various applied disciplines and in math the general form of finding the moments of some figure (a "lamina" as my book calls it) is just:
$$
\iint_R y^n \rho(x,y) dA
$$
$$
n = 1, 2, ...
$$
Related:
Intuitive explanation of moments as they relate to center of mass
First and Second Moment of Mass
Moments and Centers of Mass
Moment (physics)
Moment (mathematics)
What is the use of moments in statistics
 A: I'm not sure I understand what you're looking for here. For example, the Moment (mathematics) link you have gives a clear explanation of how the moments of a distribution characterize its shape. In physics it appears as a concrete way of thinking of a rotating object as a singular quantity to quantify its resistance to angular momentum changes. Ultimately it's just one possible way of quantifying the shape of a set of points. Assuming certain conditions are satisfied, moments of distributions are kind of like a Taylor series in that if you know all of them you can recover the original distribution. However the moments do not necessarily determine a unique distribution if certain conditions are broken. 
A: Since there have been no further answers for about a week, I'll assume I'm correct and attempt to answer here.
A little story: I ride bikes for cardio. On my way home I pass by a corn field. I rode along the perfectly straight axis of the corn field and looked at the stalks. The stalks varied in height as I rode along depending on my position along the axis. But also my distance from them thanks to perspective.
Looking at the town as I rode by, I could see that the apparent height of the objects in the distance varied with their height and my distance from them. Thus the "moment" of their apparent height as a product of their height and their distance from me.
So that defines the shape of the "distribution" of the apparent height of buildings in my town quite neatly.
It's a bit like a contour map in one dimension. If I take my moment with respect to a single axis, then I should see the "profile" of the surface with respect to that axis. If I do it in two dimensions - with respect to both axes - for a three dimensional shape, I should be able to form a contour map of the surface, tell the distribution of its mass and then by dividing by the total mass, find the coordinates of the center of mass; Where the surface would balance, on its distribution of mass over the surface.
I'm guessing that this business with the "moment of inertia" - as I have it above - "happens to be proportional" to the second moment and moments define certain key properties of some systems that define how that system behaves; E.g. if you know all the properties of a physical system you can duplicate it exactly (velocity, spin, angular momentum, etc) which some of them happen to map to moments of equations that define some of these properties.
