A writing system as a set of visible or tactile signs used to represent units of language in a systematic way. This simple explanation encompasses a large spectrum of writing systems with vastly different stylistic and structural characteristics spanning across the many regions of the globe.
Writing provides a way of extending human memory by imprinting information into media less fickle than the human brain. However, many early philosophers, such as Plato, have branded writing as a detriment to the human intellect. They argued that it makes the brain lazy and decreases the capacity of memory. It is true that many non-writing cultures often pass long poems and proses from generation to generation without any change, and writing cultures can't seem to do that. But writing was a very useful invention for complex and high-population cultures. Writing was used for record keeping to correctly count agricultural products, for keeping the calendar to plant crops at the correct time. And writing was used for religious purpose (divination and communicating with the supernatural world) and socio-political functions (reinforcing the power of the ruling elite).
In past centuries, scientists had used writing as one of the "markers" of civilization. While it is true that writing systems appear to develop in agricultural and urban cultures, by no means it is a requirement for civilization.
Because writing is so intricate there has been many explanations concerning the origins of writing, from mythological to scientific. You can read more about them in the origins of writing section.
Writing systems differ structure, stylistically, familially, geographically, and so on. Here's the several ways I categorize them:
Types: Classification according to how the system works.
Families: Classification according to "genetic" relations.
Regions: Classification according to geographical regions.
A to Z: Alphabetical listing of scripts, just for your convenience.
The study of writing systems is a very broad field, and I cannot claim that this website will reveal every detail. Far from it, in fact. I think I only cover 10% of all writing systems of the world. There are many good books out there, and you can start in my Reference section.