The difference between nombro and numero is that nombro refers to the number itself, as a concept, while numero refers purely to a label.

Ŝi havas grandan nombron da ekzemploj. She has a large number of examples. Ŝi scias la nombron de vizitintoj. She knows the number of visitors [i.e. how many they had].

Note the distinction between da and de. By contrast, numero can only be followed by de:

Ŝi scias la numeron de la domo. She knows the number of the house.

As for the other words:

amplekso — size, extent, magnitude

cifero — digit, numeral symbol

grando — bigness, size

kvanto — quantity, amount