Here is a nice *cyclic* ordering of the eight geometries:

$$\Bbb H^2\times \Bbb R,\quad \Bbb S^2\times \Bbb R,\quad \Bbb E^3,\quad \mathsf{Sol},\quad \mathsf{Nil},\quad \Bbb S^3,\quad \mathsf{PSL},\quad \Bbb H^3$$

derived from staring at Peter Scott's table of Seifert fibered geometries. The table is organized by Euler characteristic of the base 2-orbifold and Euler class of the bundle. (See his BLMS article.) The cyclic ordering also has a bit of antipodal symmetry.

I didn't come up with geometric pictures of the eight geometrics but I have thought about "icons" to represent them. (The original intent was to find eight pictures or objects suitable for a teething ring.) Here are my suggestions - I'm interested to hear what other people think/suggest.

- $\Bbb H^2\times \Bbb R$ -- triangular prism (where the triangle is slim ie ideal)
- $\Bbb S^2\times \Bbb R$ -- cylinder
- $\Bbb E^3$ -- cube
- $\mathsf{Sol}$ -- tetrahedron with one pair of opposite edges truncated
- $\mathsf{Nil}$ -- annulus with a segment of a spiral (representing a Dehn twist)
- $\Bbb S^3$ -- circle
- $\mathsf{PSL}$ -- trefoil knot
- $\Bbb H^3$ -- figure eight knot (or possibly a slim tetrahedron)

I think it is also reasonable to ask for a "prototypical" three-manifold for each of the eight geometries. Here is an attempt:

- $\Bbb H^2\times \Bbb R$ -- punctured torus cross circle
- $\Bbb S^2\times \Bbb R$ -- two-sphere cross circle
- $\Bbb E^3$ -- three-torus
- $\mathsf{Sol}$ -- mapping cylinder of $[[2,1],[1,1]]$
- $\mathsf{Nil}$ -- mapping cylinder of $[[1,1],[0,1]]$
- $\Bbb S^3$ -- three-sphere
- $\mathsf{PSL}$ -- trefoil complement
- $\Bbb H^3$ -- figure eight complement

Notice that all of the examples are either surface bundles over circles or circle bundles over surfaces, or both (i. e. products).

EDIT: In addition to the beautiful work of Zeno Rogue and Pierre Berger (linked to in the other answers) the reader may be interested in the visualisation efforts of Remi Coulon, Brian Day, Sabetta Matsumoto, Henry Segerman, and Steve Trettel. You can interact with seven of the eight here. Finally, here is a snap-shot of Sol geometry.