The regular polytopes are the analogues, in dimension higher than three, of the regular polyhedra in dimension three and of the regular polygons in dimension two. A regular polyhedron is an assembling of regular polygons, the faces, neighboring faces having an edge in common. Likewise the regular 4Dpolytopes, described by Schläfli, are assemblies of regular polyhedra, the cells, neighboring cells having a face in common.
Here are the characteristics of the six convex regular 4Dpolytopes (there are also ten stellated).
With at least three cells around each edge there are only six possibilities: 3, 4 or 5 tetrahedra, 3 cubes, 3 octahedra and 3 dodecahedra. They verify the Euler formula found by Schläfli: V + F = E + C The simplex and the 24cell are their own duals. The hypercube (also called tesseract) and the 16cell (cross polytope) are dual, the 120cell and the 600cell too. 

But how to represent a polytope? In the same manner as we do for a polyhedron: by projection on an hyperplane, what reduces the dimension by one unit; we get a representation which don't preserves all the regularity (cf perspective).
Here are representations (Schlegel diagrams) and nets of the four simplest 4Dpolytopes. Don't forget that in dimension four they are regular, in particular all their edges have same length and those of the hypercube are mutually perpendicular. For the last two it's much more complicated.



 



 
The 120cell  equivalent of the regular dodecahedron in 3D  is limited by 120 cells (regular dodecahedra), four per vertex, three per edge and two per face. This structure is an approximation of an hypersphere of 4D (just like the 12 faces of the regular dodecahedron build an approximation of a sphere in 3D).  
The puzzle on the right is a an exploded view of a projection in 3D of the 120cell; it is made up of 1+12+20+12=45 dodecahedra: only the first  at the center  is regular, the others are more or less flattened (distortions due to the projection from 4D on a 3Dhyperplane) and laidout in three successive layers.
The projections of the 30 cells orthogonal to the projection's hyperplane are the hexagonal faces (flattened dodecahedra) of the chamfered dodecahedron which contains the assembling of the 45 pieces. More precisely, the blue dodecahedron is the projection of the nearest cell (a pole of the hypersphere); the three layers (green, brown and orange dodecahedra) correspond to cells situated on a same parallel of the hypersphere; the dodecahedra flattened into grey hexagons correspond to the cells situated on the equator. Two cells symmetric with respect to the equator have same projection. Want to build your own (paper models or 3D printing)?

 
Thanks to Arnaud Chéritat who kindly sent me the data for this animation and to Nicolas Hannachi who transformed them with Mathematica to make them usable with LiveGraphics3D. 
There are only three regular convex polytopes  with respectively d+1, 2d and 2^{d} hyperfaces (the three first rows of the table)  in each dimension higher than four; the first, corresponding to the tetrahedron, is self dual, the two others, corresponding to the cube and the octahedron, are dual.
Remark: the characteristics of the nDhypercube can be deduced from the coefficients of the polynomial (1+2x)^{n }, for n=4 (1+2x)^{4} = 1 + 8x + 24x^{2} + 32x^{3} + 16x^{4}
In this video (YouTube  7'20) Oliver Knill presents the six 4polytopes and the three 5polytopes.
references: 
• To understand and visualize the 4th dimension view the videoanimations proposed by dimensionsmath (2 hours of beautiful mathematics!); there you'll also learn  with an elementary proof  what a stereographic projection is.
• Two other approaches: 4D Visualization and La quatrième dimension (Micmaths on YouTube, in French), a set of four videos (définition  représentation  curiosités  hypercube) by Mickaël Launay • The 4Dpolytopes on Wikipedia (French version) • To play with 4Dpolytopes  at least with their projections in 3D  you need Stella4D, the outstanding program by Robert Webb. • The Tesseract by Alex Bogomolny (with interactive applets) • Le 120 (hyperdodecahedron) by Arnaud Chéritat (in French)  build your own (paper models or 3D printing) • "120 Cell Explorer" allows you to discover and handel the 120cell • Die Platonischen Polychora by Marco Möller (in German) • Regular 4d Polytope Foldouts by Andrew Weimholt • Images for Mathematics: 4D geometry (with animations) • Barn Raisings of FourDimensional Polytope Projections by George W. Hart • FLATLAND : geometric fiction from Edwin A. Abbott (1884) which presents a two dimension world (it inspired at least two animation movies) 
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convex polyhedra  non convex polyhedra  interesting polyhedra  related subjects  october 1999 updated 31072022 