Extra Dimensions (in String Theory) - What does it mean? I have been reading a lot about string theory and the necessity of extra dimensions (particularly as visualized in Calabi-Yau spaces), as "curling-ups" in our apparently 3-dimensional (or 4-dimensional, including time) world. I do not understand, though, how all these "extra dimensions" are actually adding dimensions to our world.
Even if these little multi-dimensional curls are everywhere and too tiny for us to see, how is it still adding extra dimensions? I mean, if our eyes were sharp enough, couldn't one still specify their location according to an $x,y,z$ axis? They will be a certain amount to the "left," a certain amount "high," and a certain amount front or backward (etc). It's still length, width and depth. 
What am I missing here?
 A: Take an antlike entity  on a two dimensional surface, it is like an ant but it has only two dimensions moving within  the (x,y) plane. One can pile up an infinity of (x,y) planes with ants in their surface when a z direction is hypothesized by an Einstein of an ant.
In this example the ant could disappear from its plane, with a perpendicular motion  and appear on other planes. 
If we hypothesize a fourth space dimension for our three dimensional universe , we could disappear into the perpendicular direction into another three dimensional volume.

They will be a certain amount to the "left," a certain amount "high," and a certain amount front or backward (etc). It's still length, width and depth.

The example of a two dimensional ant should make you understand that different surfaces should exist when adding the third dimension.  For the three to four dimensions it is different volumes. And also funny shapes, part within our volume and part outside it would really be confusing us.
String theories need extra dimensions so as to be able to embed the standard model of particle physics into the vibrations of the strings. The standard model is an encapsulation of practically all the data we have for the quantum mechanical framework of particle physics. 
As people do not in reality disappear into other extra  dimensional volumes, theorists needed to make the extra dimensions very tiny, to agree with the experimental fact that only in fairy tales people disappear and reappear. The compactified 
curled up extra dimensions  still fulfill the need for embedding the standard model, and also the fact that no extra dimensions have been observed in all our particle experiments either.
In the example of the ant, if one compactified the third ,z, dimension, there would be no danger of it sliding around surfaces not existing in its original world, because it could not fit into the curled dimension.
A: I'll try to throw in my understanding of this, which is unfortunately limited to only having visited the String theory-lecture two times, but let me try:
Strings in extradimensions are often portrayed as swinging strings. Let's stay with that analogy. Any direction a string can vibrate into, like the x and y plane, we call a dimension, or degree of freedom. X and Y coordinates are not limited in any way, we call them "macro" dimensions.
Now there are reasons to believe that other types of degrees of freedom might exist. That would go into group theory and shall be omitted here. But the essential idea is that there might exist coordinates which are limited in their value, and periodic in nature.
Just like an angle-like variable that goes round on a ring. This leads to the often quoted cylinder analogy, which is supposed to try to understand this concept better, where you take the long axis of the cylinder to be one of the x-y-z axis, and the cylinder mantle now consists of those new coordinate rings.
Each ring there is a degree of freedom, and the coordinate that lives on that ring is independent of the coordinate in the next ring. In this sense we added a 'rolled up' dimension into which the string can swing into, which is not visible on a macro-scale.
Other than that I fear that words fail us here, and we'd need to go into the math to develop a deeper understanding of all that. 
A: Since you have asked a second question through your answer I will now respond to the point raised there (and leave it to the mods to merge your two "questions" together). It is clear you have thought carefully about this and you deserve an answer to your doubts, so here goes... 
The reason for your confusion is that you are trying to represent four dimensions on a three- (in fact it's really two-) dimensional piece of paper. Let's look at your pictures and see why.
The three dimensional cube you drew: put a point somewhere inside of it. Why can't I describe it with just the $x$ and $y$ coordinates? Well as you know, we also need the depth "into" the page, the $z$ component, because we can hold $x$ and $y$ fixed and vary $z$ to arrive at distinct points, agreed? But if I draw it on a 2D piece of paper (without an isometric perspective) then this would not look so clear. How would you distinguish points with the same value of $x$ and $y$ and different values of $z$ (again assuming you do not take advantage of isometric axes to give the illusion of three dimensional space)??
The same applies to your final picture - the "four dimensional cube" that you drew on a three dimensional set of axes connected by grey lines that indicate the fourth axis (that is then rendered on a two dimensional planar monitor) suffers the same issue. It appears that you can describe your point inside the figure using just $x$, $y$ and $z$ but this is the same fallacy that I tried to clarify in my previous paragraph; in fact, if you hold $x$, $y$ and $z$ fixed, each distinct value of $w$ corresponds, by definition, to a distinct point in this space. In other words, I can move your point along grey lines in such a way that the $x$, $y$ and $z$ values remain unchanged (even if you can't see this in your figure) and the $w$ value varies. The problem is that all of this is projected onto three axes that give the illusion that you don't need to the fourth axis. 
A: I'm writing an answer because I need to include pictures and I don't know how to with comments...
I guess I need pictures to understand this. Words are not doing it for me thus far, I guess. 

Looking at this picture, the faint gray lines connecting the cubes are the "fourth" dimension. 
But I still don't understand how we really need this "fourth dimension" to describe where the points on the gray lines would fall. Why wouldn't you be able to describe them just using the x, y, z axes. Let's look at the picture below. 

Let's say there is a little red dot (look at the fourth picture). One could imagine it at 0-x, 0-y, -2 z. If I follow the "fourth dimension" lines to the purple dot I drew, it seems to me the z coordinate would stay the same; you'd just have to change the x and y. The new cube and the gray lines have moved up and to the right, but they could still be described using the z plane.
It seems to me all the extra dimensions of a "curling-up" produced by the Calabi Yau manifolds can still be described using only 3 dimensions!! 
