There are three basic forms of writing systems: logographic systems, which use symbols to represent whole words; syllabary systems, which use symbols to represent syllables; and alphabetic systems, which use symbols to represent units of sound. Logographic systems are the most intuitive from the perspective of a society on the cusp of developing writing, and thus they tended to be the first to arise. They also are most logical next step from the marking system used by merchants and accountants. (Incidentally, the fact that Chinese is a logographic system is one piece of evidence for its independent invention - it bears little resemblance to the cuneiform of the Sumerians and is of the type expected for an early writing system.)
Logographic systems, while being highly intuitive at first, quickly become extremely cumbersome. They are difficult to learn and give relatively few pronunciation cues. Moreover, they require the invention of a new syllable every time a new word is needed, and they make compound words and complicated syntax much more difficult to write. Finally, the complicated pictorial symbols must be rendered almost perfectly in order to be legible, which makes writing a time- and energy-consuming process.
The second system, the syllabary, is much rarer, since it is an intuitive system only for a few languages. It is used in one variant of Japanese and was developed by an extremely intelligent Cherokee, Sequoyah, for use in writing and recording his native language. His system, based very loosely on English (at most he borrowed a few forms), became so successful that the formerly illiterate Cherokee tribe began publishing newspapers and books in their own language.
The third system of writing, the alphabetic system, is the most difficult to invent and the easiest to use. Many linguists believe it was invented only once, by the Phoenicians, and then spread or adapted to other languages. The system seems counterintuitive at first, since its most basic units do not correspond to anything meaningful in speech, but rather to an isolated sound. However, the system uses the power of infinite combination to achieve its success; whereas Chinese characters might take years to learn, the standard Roman alphabet often takes only a few months for children to memorize. Moreover, when each letter represents a certain sound, pronunciation is more easily inferred from the structure of a word (though English pronunciation sometimes leaves speakers confused). Finally, markers such as umlauts serve to increase the power of a system by more carefully delineating the pronunciation of certain letters.
In general, when logographic, syllabary, and alphabetic systems compete, the alphabetic system will tend to dominate because it can express the most thoughts most efficiently. Languages such as Japanese and Chinese will probably eventually be outcompeted because English - or another language based on an alphabet, most likely the Roman one - is so much easier to use. (This is not to say that English is easy, only that the alphabetic system is most efficient.)
