The Arabic Writing System الخَطّ العَرَبيّ
The use of writing seems so obvious to us that it is difficult to imagine that it is a relatively recent invention. Writing was created in more than one place on earth, including ancient Mesopotamia (today's Iraq, as early as 3100 BCE), ancient China (perhaps 2000 BCE), and the native Americans (relatively late, perhaps the second century CE). Writing also appeared in ancient Egypt around 3000 BCE. Scholars are still divided as to whether writing in Egypt was an independent invention, or whether the idea of writing came to Egypt from Mesopotamia.
The writing system used in ancient Egypt, called "hieroglyphics," was complex. Some signs stood for a single consonant, some stood for two consonants, some stood for three consonants, and some stood for entire words. Some time around 1500 BCE, an unknown genius, living somewhere in the general area of Syria-Palestine, simplified the system to its essence, creating an actual "alphabet" أبَجَدِيّة. An alphabet is a writing system in which each symbol stands for only one sound, and each sound is represented by only one sign.
We only have a few traces of this earlier alphabet. It came into the full light of history when it was adopted by a people called the Phoenicians الفينيقيّون. The Phoenicians lived in what is today's Lebanon then spread further west of the Mediterranean in Carthage (today's Tunisia). The Phoenicians spoke a Semitic language. We have a number of Phoenician inscriptions, going back as early as roughly 1000 BCE. The Phoenician writing system was adopted by several peoples in the Ancient Near East, including the ancient Hebrews and Aramaeans.
The Phoenicians were a mercantile people so they carried their alphabet around the world. We have Pheonician inscripions from as far away as France and England. The writing system they used was eventually adopted by the ancient Greeks, perhaps in the 9th century BCE.
In the Phoenician writing system, each symbol represented a consonant. Vowels were not written out. In Semitic languages consonants carry more weight than vowels do because vowels are largely predictable - hence the omission of the vowels. This is not true for Greek, an Indo-European language, where the vowels are as important as the consonants. The ancient Greeks thus took the additional step of creating symbols to indicate vowels. With these new symbols, the Greek writing system was passed on to the Etruscans, who then passed it on to the Romans, and it then eventually became used for other languages such as English.
Although writing was invented several times, the alphabet was apparently only invented once. All alphabetic writing systems that we see today are descendants of this one alphabet, no matter how different they look. Just looking at, for example, English, Russian, Arabic, and Mongolian, it is hard to imagine that their writing systems are directly traceable to the Phoenicians.
In the case of Arabic, Arabs living in the Arabian Peninsula borrowed their writing system from a people called the Nabateans. The Nabateans were Arabs who lived around the time of Christ in the city of Petra, in today's Jordan. The Nabateans themselves had adopted the writing system of an ancient people called the Aramaeans, who had borrowed it from the Phoenicians. In this long process, the shapes of the letters changed so much that Arabic bears hardly any surface resemblance at all to ancient Phoenician.
As was mentioned above, in the course of the Islamic Conquests, Islam was carried out of the Arabian Peninsula. The Arabs carried their writing system with them, and it was adopted to write many languages, some of them related to Arabic and some not. Thus today Arabic is used to write Persian and Urdu, both Indo-European languages; up until 1928 it was used to write Turkish, an Altaic language. There are sporadic cases of it being used for Spanish, Afrikaans, Polish, Hungarian, Hebrew, even Latin.
In addition to the eminently practical uses to which the Arabic writing has been put, it is also used for aesthetic purposes. Calligraphy, a Greek word meaning "beautiful writing," is one of the most highly developed art forms in Islam. This was true for classical Islamic civilization, and it is true today, with contemporary calligraphers using all modern tools, including the computer and 3-d modeling, to produce stunningly beautiful works of art.