Many names are constructed from Greek prefixes for the number of sides and the root -hedron meaning faces (literally meaning "seat"). For example, dodeca-, meaning 2+10, is used in describing any 12-sided solid. The term regular indicates that the faces and vertex figures are regular polygons, e.g., to distinguish the regular dodecahedron (which is a Platonic solid) from the many dodecahedra. Similarly, icosi-, meaning 20, is used in the 20-sided icosahedron, illustrated at right. (Note: the i turns into an a in this word only; elsewhere it stays i.) Following this pattern, some authors call the cube the hexahedron. The term -conta- refers to a group of ten, so a hexecontahedron has 60 sides.
Modifiers may describe the shape of the faces, to disambiguate between two polyhedra with the same number of faces. For example, a rhombic dodecahedron has 12 rhombus-shaped faces. A pentagonal icositetrahedron has 24 (i.e., 20+4) five-sided faces. The term trapezoidal is standardly used to refer to "kite-shaped" quadrilaterals, which have two pairs of adjacent sides of equal length (and so are not trapezoids by the modern American definition which requires that two opposite sides be parallel). Thus a trapezoidal icositetrahedron has 24 such faces. (This usage is not as odd as it may first seem; a British definition of trapezoid is "a quadrilateral figure no two of whose sides are parallel" --- Oxford English Dictionary.)
The term -kis- refers to a the process of adding a new vertex at the center of each face and using it to divide each n-sided face into triangles. A prefix corresponding to n standardly preceeds the kis. For example, the tetrakis cube is derived from the cube by dividing each square into four isosceles triangles. A pentakis dodecahedron is based on the dodecahedron, but each pentagon is replaced with five isosceles triangles. (In these cases, the tetra- and penta- are redundant and in most cases kis- alone would suffice.)
Numerical modifiers like pentagonal or hexagonal can refer not only to the shape of individual faces, but also to a base polygon from which certain infinite series of special polyhedra can be constructed. For example the pentagonal prism and hexagonal prism are two members of an infinite series. Related infinite series are the antiprisms, and the dipyramids and trapezohedra.
The term snub can refer to a chiral process of replacing each edge with a pair of triangles, e.g., as a way of deriving what is usually called the snub cube from the cube. The 6 square faces of the cube remain squares (but rotated slightly), the 12 edges become 24 triangles, and the 8 vertices become an additional 8 triangles. However, the same process applied to an octahedron gives the identical result: The 8 triangular faces of the octahedron remain triangles (but rotated slightly), the 12 edges become 24 triangles, and the 6 vertices become 6 squares. This is because the cube and octahedron are dual to each other. To emphasize this equivalence, it is more logical to call the result a snub cuboctahedron but it may take a while for this name to be widely adapted. Applying the analogous process to either the dodecahedron or the icosahedron gives the polyhedron usually called the snub dodecahedron, but better called the snub icosidodecahedron.
There are four Archimedean solids which each have two common names:
The term stellated almost always refers to a process of extending the face planes of a polyhedron into a "star polyhedron." There are often many ways to do this, resulting in different polyhedra which are not always well distinguished with this nomenclature. For examples, see the 59 stellations of the icosahedron. But be aware that some authors have incorrectly used the term stellate to mean "erect pyramids on all the faces of a given polyhedron," and a few mathematicians have suggested a stricter definition of stellate based on extending a given polyhedron's edges rather than faces.
The term compound refers to an interpenetrating set of identical or related polyhedra arranged in a manner which has some overall polyhedral symmetry.
The term pseudo shows up in two "isomers" which are rearrangements of the pieces of a more standard polyhedron.
Names for many of the nonconvex uniform polyhedra and their duals have been in flux. The two books by Wenninger which illustrate these polyhedra list names largely due to Norman Johnson. The names evolved slightly between the two books [1971, 1983] and since. I have incorporated his most recent naming suggestions at the time of this writing.
Crystallographers use a slightly different set of names for certain crystal forms.
For a systematic method of naming a great many interesting symmetric polyhedra, I like John Conway's notation.
Answer: This and this. (Joe Malkevitch showed me the infinite families that these members of.)