Differentiable manifold has 8 or 9 syllables and it is easier to say smooth set, which only has 2 syllables. And that is what one is. It is a set with a bunch of coordinate charts that work smoothly together.

Typically you cant get the whole set on one coordinate chart so you have several overlapping charts

that is like you cant get the whole earth on one square map, but you can plaster maps all over the earth so you have overlapping coverage.

On every patch of surface there is some map that is good at least on that local region.

the typical set used to represent space in LQG is the "3-sphere" where the surface of a balloon is a 2-sphere and you have to imagine going up one dimension. a local chart looks like regular 3-D graph paper or familiar euclidean 3-space

Only thing is we ignore the geometry you might have thought we had when I said 3-sphere. If we were thinking of the 2-sphere balloon as an analogy, the air is out of the balloon and it is crumpled up and thrown into your sock drawer. it has no shape. In the same way, by analogy, the 3-sphere has no shape. It is just a set of points, without a boundary, that has been equipped with an adequate bunch of coordinate maps

the "smooth" part is that whereever the charts overlap if you want to start on one map and find the corresponding point on the other map, and do a whole transference thing that remaps you from one to the other, well that

remapping (from one patch of 3-D graph paper to another) is smooth. that is to say differentiable, as in calculus, you can take the derivative as many times as you want. In other words the coordinate charts are COMPATIBLE with each other because whenever you remap between two that overlap you find you can DO CALCULUS at will on the function taking you from one to the other. this is an example of a technical condition that basically doesnt say very much except that we wont have nasty surprises when we get around to using the charts. The charts are smoothly compatible with each other.

The idea of a differentiable manifold was given us by George Riemann in 1854 when he was trying to get a job as lecturer at Göttingen and had to give a sample lecture, and it is actually SIMPLER than euclidean space because it does not have any geometry! Euclidean space has all kinds of rich structure immediately availabe, like you can say what a straight line is and you can measure the angle between two intersecting straight lines!

what we have here is a SHAPELESS SMOOTH SET and you cant do any of that. It has the absolute mininum of structure for something that can serve as a useful model of a CONTINUUM.

this is why it was a good idea of Riemann, because it is simpler and less structured than Euclidean space and so it is more able to adapt to the wonders of the universe. mathematics was changed very much in 1854.

George Riemann lived 1826 to 1866.

Here is his 1854 talk, in full:

http://www.ru.nl/w-en-s/gmfw/bronnen/riemann1.html
I think what he called a "stetige Mannigfaltigkeit" here in this talk we would call a smooth manifold. But thereafter the name "differenzierbare Mannigfaltigkeit" became prevalent and is what we call differentiable manifold.