Take a naive interpretation of regular polyhedra:


All vertices (including epsilon ball) congruent


All edges congruent


All faces congruent


We can now find interesting families by removing one requirement. For example the uniform polyhedra have all vertices and edges congruent, but not all faces, and their duals have faces and edges congruent, but not vertices. 


Are there examples, or interesting families, of polyhedra where every pair of faces is congruent and every pair of vertices, but not every pair of edges?



Not quite regular polyhedra
It turns out that these polyhedra that have congruent vertices and faces have a name. They are the *Noble Polyhedra*. If one insists that they also be convex the Noble polyhedra are the regular polyhedra plus the [disphenoids](http://en.wikipedia.org/wiki/Disphenoid) mentioned in Douglas Zare's answer. 


When one allows intersecting faces, however, new collections turn up, such as the stephanoids, originally studied by Max Brüker:


Max Brückner *Uber die gleichecking-gleichflachigen, diskontinuierlichen und nichtkonvexen Polyheder*  

Nova Acta Leop. 86(1906), No. 1, pp. 1 – 348 + 29 plates.  

[Images of the plates with pictures of the models.](http://bulatov.org/polyhedra/bruckner1906/index.html)


These shapes are also discussed and further developed by Branko Grünbaum:


Polyhedra with hollow faces  

*Proc. NATO-ASI Conf. on polytopes: abstract, convex and computational*, Toronto 1983, Ed. Bisztriczky, T. Et Al., Kluwer Academic (1994), p 43-70.


Grünbaum's constructions do use generalisations of the definition of polyhedra. For a thorough discussion of these (including having polygons return to the same vertex, and coplanar faces) see the following paper, which also has a discussion of Noble Polyhedra. 


Grünbaum, B. Are your polyhedra the same as my polyhedra?  

*Discrete and Computational Geometry: The Goodman-Pollack Festschrift* B. Aronov, S. Basu, J. Pach, and Sharir, M., eds. Springer, New York 2003, pp. 461 – 488.  

<http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf>


Interestingly the classification of all Noble Polyhedra is still an open problem.


