A polytope is a catchall term for flat shapes regardless of how many dimensions it has(2d 3d 4d etc). The distinction that they must be flat eliminates things like spheres, cones, circles, etc. This is an arbatrary distinction but it does give polytopes certain properties that are interesting as well as leading to other groups within it that are very interesting like the star polyhedra or the catalan solids.

The primary kind of polytope that I at least care about and read about is 3 dimensional polytopes known as polyhedra(more on that later), but there's also the familiar 2 dimensional polytopes aka polygons. Polytopes are always made up of polyhedra of the dimension below. While I'm not sure if it counts 1 dimensional polytopes would just be lines so polygons are shapes made of lines, and polyhedra are shapes made out of polygons.

Now I'm not as familiar with these but I can give an overview. From what I've read about I really only know about 4 dimensional polytopes which are called polychora. The term dimensions in things like science fiction or video games it's like an alternate reality but mathematically speaking it doesn't imply that at all; it just means that there's another axis, like with 2d coordinates there's two axis, the x and y axis, and in 3d there's 3, x, y, and z. In 4 dimensions there's x, y, z, and w. Talking about 4 dimensional objects in this manner doesn't mean that a 4 dimensional space exists it's just exploring the properties of what 4 dimensional objects would have. There is no unique name for polytopes in 5 dimensions or above, they're just called 5-polytopes, 6-polytopes, etc. I also don't look much into polytopes above 4 dimensions or even 4 dimensional polytopes themselves but just so you know that's how it extends to higher dimensions.