The point I expected this page to say that it doesn't really mention is that there's a fairly clear progression in the development of writing. Early writing systems all seem to start with some sort of pictography. This runs into the problem that not every word can be represented with a clear pictogram, and the solution ends up being some sort of phonetic punning mechanism (i.e., the rebus principle). Even Chinese hanzi evidences these sort of mechanism, as the vast majority of its characters are based on combining a phonetic component with a semantic component.

After you start phonetic punning, most writing systems drop the pictographic portion and become solely based on phonetic constituents (Chinese being the major exception, although Japanese did evolve the next step into kana). These constituents are usually roughly based on a consonant-vowel pair and writing systems in this phase are called syllabries, but they're often not really full syllables, and there are tricks for representing more complex syllables (for example, Mayan uses the principle that the syllable pattern CVC is represented with CV-CV, where the vowel is the same in both characters.).

Now, in Phoenician or thereabouts, an interesting turn happened. Because of the nature of Semitic languages, the syllabry form dropped all of the vowels and just represented the consonants (what's now termed an abjad). The Indo-European speakers (principally Etruscan and Greek) who borrowed from this alphabet repurposed some of the letters which corresponded to missing consonants (such as the glottal stop in 'aleph) to represent vowels instead, as vowels are rather more important in those languages. Other languages that borrowed from abjads indicated vowels with systematic vowel marks, such as Devanagari or Canadian Aboriginal Syllabics; these are known as abugidas. Virtually every alphabet, abjad, and abugida known to exist is believed to derive originally from this first "wrong" turn when developing syllabaries.

An interesting point when discussing the history of writing systems is the development of the Cherokee script. The Cherokee script was invented by Sequoyah, who developed it entirely on the basis that he knew that the white settlers had a writing system that could be reliably read and a copy of the Bible but without any other knowledge of writing. He originally attempted to develop a mapping of word => character, but realized that the mapping is too complex, especially since the text he had got by with much fewer characters. Instead, he chose a syllabary, borrowing some of the letter forms from Latin script but not their phonetic mappings (which is why ᏣᎳᎩ is actually pronounced "tsalagi").