Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow may have a better knowledge of the literature here.
A monohedral polyhedron is one whose faces are all congruent.
We have a classification of all convex isohedral (face-transitive) polyhedra, consisting of 30 classes of assorted finite polyhedra and infinite families. See Wolfram Mathworld for a list.
In the process of writing this answer, I was trying to find instances of convex polyhedra which are monohedral but not isohedral, and struggled to find a classification of such shapes or even a list of known instances with an outline of which cases remain open.
The instances I know of beyond the isohedral polyhedra given above:
The pseudo-deltoidal icositetrahedron, the dual of the pseudorhombicuboctahedron. In footnote 46 on page 185 of Advances in Discrete and Computational Geometry: Proceedings of the 1996 AMS-IMS-SIAM Joint Summer Research Conference, Discrete and Computational Geometry, it is remarked that no other convex non-isohedral monohedra are known with non-triangular faces (or at least, none were known in 1996 - but see the next two bullet points). (Here is a Google Books link to the relevant section of the previous source.)
However, in a result apparently unknown to the above source, Ed Pegg provides what seems to be a counterexample in this math.SE question; manually cutting out and folding the net, it does indeed seem to fold into a convex polyhedron.
In the article The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra (Grünbaum, B., 2010), the author includes the rhombic icosahedron and the Belinski dodecahedron, and cites Belinski as proving that these are the only convex monohedra with centrally symmetric faces (such polyhedra termed isozonohedra) not already listed among the isohedra.
In David Eppstein's paper On Polyhedral Realization with Isosceles Triangles (link to arXiv abstract), three infinite families of convex monohedra not given above are listed, with isosceles triangle faces: one consisting of a heightened antiprism with pyramids glued to each large opposite face (called a "gyroelongated bipyramid"), a variant of the previous shape in which two halves are rotated about a skew hexagonal cross-section (called a "twisted gyroelongated bipyramid"), and a shape called the "biarc hull" which is related to a sphericon. They cite the paper Some New Tilings of the Sphere with Congruent Triangles by Robert Dawson, most of which seem not to translate to monohedral polyhedra but which I haven't checked fully.
The answers to this math.SE question, which provide monohedral polyhedra combinatorially equivalent to the icosahedron but with non-equilateral faces. (The scalene case is not always convex, but can be made so with small distortions to the angles of the triangles involved.) The book Advances in Discrete and Computational Geometry mentioned above goes into more detail about the possibility of such polyhedra.
Given the apparent lack of a classification of more restricted classes of polyhedra, I expect that no complete classification exists. However, I'd like to better understand the scope of which examples and impossibility results are known, as I haven't been able to find much in the way of a definitive source tackling this question. I'm hoping this question and its answers can at least serve as a better overview of known convex monohedra than the scattered state of information that seems to exist on the problem at present.