Best not let any linguists hear you say "representing concepts to representing sounds". All languages represent sounds as far as I've ever heard from them, and the idea of "representing concepts" is regarded as a remnant of some very broken ideas. :-)
As far as I (a non-linguist, although I've read a bit) know, there are roughly three ways of writing: alphabets, syllabaries, and whatever Chinese is (logosyllabic?).
Alphabets have one grapheme per (roughly) phoneme and have(AFAIK) developed once: Proto-Siniatic, which led to Semitic (Hebrew, Arabic, etc.), Phoenician (which developed into Greek, Latin, etc.) and a few others. Every other alphabet is a direct or indirect descendant. They typically have <50 graphemes.
Chinese is kinda-sorta syllabic, but the language has a great many one syllable words which have a grapheme mapped directly to that use. On the other hand there's things like "coral" (IIRC) which is two syllables and is written with two characters each of which are not used anywhere else. It, its descendants, and any other similar languages if there are any, has some many thousands of graphemes.
Syllabaries have one grapheme per syllable and are the most common form of writing, to the extent that it's pretty clear that they're the normal version of human writing. They typically have a few hundred graphemes.
Mayan and cuneiform are a couple of weird cases. They (AFAIK) are mostly syllabic, but with some logographic-ish parts like Chinese. But the number of graphemes are pretty firmly in the syllabary range.
Tl;dr: Writing is weird and the writing I'm doing now is very much so.