Riemann makes clear that an n manifold looks locally like Euclidean n space, in that points (elements) can be fully described by n continuous functions. As Fresh said, it is not clear whether he is thinking of charts or embeddings, but he is using only n coordinates near every point. If we imagine the 2-sphere embedded in 3-space as usual as x^2+y^2+z^2 = 1, then we can cover the sphere by 6 pairs of open hemispheres n(x>0, x<0, y>0, y<0, z>0, z<0), on each one of which two of the coordinates, i.e. either (y,z) or (x,z) or (x,y), will provide such local coordinates.
Thus we can think of the sphere as embedded by the three functions x,y,z, of which we choose an appropriate two near each point, or we can think of the sphere as covered by those 6 charts, on each of which we only consider two functions, which give a homeomorphism of that chart, and which are smoothly related on overlaps.
Conversely, given a (finite) covering by charts, we can presumably extend each chart homeomorphism to n globally defined smooth functions, and then use all of them to embed the manifold in a big product space. Hence there is little difference between the two approaches, at least for compact manifolds. Riemann moreover speaks of infinite dimensional manifolds, even uncountably infinite dimensional ones, well before Cantor made this precise.
The more Riemann I read, the more I see that he anticipated so much later mathematics; it is almost as if most later mathematicians, even great ones, just took Riemann and worked out the details.
The moral for those of us wanting to do research, or just to understand mathematics, is to try to read the great ones, as well as we can, and think about questions that arise.
As something I just learned by reading a few pages of Riemann's lecture, we recall that Euclid "proved" the SAS congruence theorem by appealing to transformations of the plane carrying one triangle to another with the same measurements. Riemann remarks that of course this sort of isometric transformation of figures is possible also on any surface of constant curvature, a fact I had never realized clearly, although it seems obvious on the sphere.
He makes another more cryptic remark that such transformations are possible even without "bending", on a surface of positive curvature, but not on one of negative curvature. This seems plausible for embedded surfaces, but ....????? Another example of a brief remark by Riemann that may repay a lot of thought.ot on