Moment if inertia is the rotational analogue of mass. But I can't get the idea of the term "moment of area". What does it mean?
I agree that this question seems to be about the "second moment of area", but it's a bit more than "how wide" the object is. For example, in statistics, standard deviation has the same units as the quantity whose distribution is being studied, but the second moment of area does not have the units of a width, or even an area, it has the units of distance to the fourth power. In particular, if you take two planar objects with the same shape and made of the same substance, but all linear scales are double for one versus the other, then the second moment of area is multiplied by 16, which is surprising until you think about the fact that the concept is designed for planar objects of the same thickness and made of the same substance, so what goes up by 16 is the width-squared-weighted area of those objects.
"Moments" take some quantity and weight it in a range of important ways, such that if you had the complete list of all the moments, you would understand the shape of that object. So the first moment of area is just the area of the object, like the first moment of the mass is just the mass. The second moment of area takes the area and weights it by the square of the distance to some axis. The purpose of doing that weighting is to understand how the area will act when you either put it under stress (as when you try to bend a column in architectural engineering) or try to make it rotate (the more common application in physics).
Since you mention the moment of inertia, let's assume you have in mind the physics application. Then the idea is, imagine a collection of planar objects, all with the same shape, but different areas and masses. The moments of inertia of all those objects are different, but since they are all the same shape, there must be some aspect of the moment of inertia that is the same. That aspect is the second moment of area, which is the quantity that needs to be multiplied by the mass and divided by the area to get the moment of inertia of every one of those planar objects. In short, it is the moment of inertia per surface mass density of those planar objects. When one uses it, one is saying that one understands that the moment of inertia will be proportional to the surface mass density, and what one wishes to understand is how the shape affects the moment of inertia, not how the surface mass density affects it. Then you calculate the second area moment, multiply by the more obvious surface mass density (which is a function of the material and thickness of the objects), and you get the mass moment of inertia that you need to understand the response to torques.
In general, a moment is a quantity that describes the shape and position of something. In statistics, the mean and standard deviation are the first and second moments of a distribution; they are numbers that tell the reader approximately where the distribution is located (the mean) and approximately how wide it is (the standard deviation).
In physics, we have similar notions for the location and shape of an object, which we call the moments of area. The first moment of area tells us where an object is (i.e. the location of its center of mass). When we analyze a rigid rotating body, we usually do so in a frame that places this at the origin, which is why it is usually irrelevant in calculations. The important quantity for physicists is the second moment of area, which, much like the second statistical moment (the standard deviation), tells us how wide an object is. You might recognize this quantity (with proper unit conversion) as the typical moment of inertia.