I learned the Schläfi symbol notation from Coxeter’s “Regular Polytopes”. I will proceed by example:

{3} denotes an equilateral triangle and {4} denotes a square.
{3, 3} denotes a tetrahedron bounded by triangles meeting at vertices by threes.
{4, 3} denotes a cube bounded by squares meeting by threes.
{3, 4} denotes an octahedron bounded by triangles meeting by fours.
{3, 5} denotes an icosahedron bounded by triangles meeting by five’s.
{5, 3} denotes a dodecahedron bounded by pentagons meeting by threes.

Note that reversing the numbers yields the dual figure; that pattern persists in more general cases below. Each of the above can be viewed as tiling the sphere. The general pattern so far is that curly braces with n elements denotes an n+1 dimensional polytope whose n dimensional faces meet at n−2 dimensional ‘edges’. (They also meet at n−1 dimensional places, but always by twos.)

{3, 6}, {4, 4}, and {6, 3} each tile the flat plane.
{5, 4} tiles the hyperbolic plane and my PostScript program draws that tiling on the Poincaré disk, and a movie of same. See this for more.

{4, 3, 4} tiles flat 3 space. {4, 3} denotes a cube and {4, 3, 4} denotes packed cubes, four about shared edges. In these constructions an n dimensional polytope meets neighboring polytopes across an n−1 dimensional polytope which is a face of each of the first two. Several congruent polytopes may ‘surround’ and meet at an n−2 dimensional polytope and the count of these provides the values in the Schläfi symbols. Such meetings are the bones of curvature; the space is flat elsewhere.

Aristotle thought that {3, 3, 5} tiled space, but he was wrong! Here is an applet to show {5, 3, 3} projected onto the screen. You can rotate it in 4-D.

There are two closely related spaces associated with each of these designations:

• The cube, {4, 3} has flat faces with no curvature. Even the edges can be unfolded to become flat. Only the vertices have intrinsic curvature and each of them has a 90 degree deficit which is counted as positive curvature. The eight vertices of the cube together have 720 degrees or 4π of curvature. All simply connected objects in 3 space have the same total curvature of their surface.
• The other view of {4, 3} is obtained by imagining a cube inscribed in a sphere. The edges and vertices of the cube are projected onto the sphere from the center. Now we have a tiling of the sphere with ‘spherical squares’. Likewise if we inscribe an icosahedron {3, 5} in a sphere and project to edges to the sphere’s surface, the sphere is tiled by spherical triangles. The sphere’s curvature is uniformly distributed.
The same two ways of considering tilling applies to negatively curved spaces. For this note we shall henceforth imagine continuous curvature, in fact constant curvature. I think that there is a connection to homotopy theory.

For uniform curvatures the space is Euclidean in the small. If, in a 3-D tiling, we consider a small sphere about a vertex, the sphere is itself tiled. Being embedded in a Euclidean space, the sphere must itself have a positive curvature. This rules out {4, 3, 6} as a 3-D tiling for the said sphere would then be tiled by {3, 6} which does not have positive curvature. There are tedious elementary formulae indicating whether the tiled space is positively or negatively curved.

The tiling {4, 3, 5} tiles the vertex spheres as {3, 5}. The vertex of that tiling is thus like an icosahedron and there are 12 edges meeting that vertex.

See this for more about the geometry of curved spaces.