In the beginnings of Islam, the Quran was mostly recorded in the memory of the Huffaz. After witnessing the unreliability of such a form of transmission, mostly because of the untimely death of many of those Huffaz in battle, it was decided to record it in written form.
Given the sacred nature of the word of Allah, the book would be made with great attention to quality and readability. Given Islam's taboo against pictural representation, however, drawings could not be used to enjolivate the book, as was done in the Christian world. Neither was it allowed to decorate mosques, for that matter. Thus, the art of calligraphy became very important in the Muslim world, and still today it is a major art; calligraphers are held in great esteem. The aesthetic of their art, which allows for the teaching of the Quran, is an unifying aspect of Islam.
After the definitive fixing of the arabic script around 786, by Khalil ibn Ahmad al Farahidi, many styles were developped, both for the writing down of the Quran and other books, and for inscriptions on monuments as decoration.
The first of those to gain popularity was known as the Kufic script; it was angular, made of square and short vertical strokes, long horizontals, and bold, compacts circles. It would be the main script used to copy the Quran for three centuries; its static aspect made it fit for monumental inscriptions, too. It would develop many serifs, small decorations added to each character.
More often used for casual writing was the cursive Naskh script, with rounder letters and thin lines; with perfectionning of its writing techniques it would come to be preferred to Kufic for copying the Quran.
Developped later, the Thuluth would take on from the 13th century the ornamental role devoted to the Kufic script. Thuluth meaning one third, it is based on the principle that one third of each letter slides downward. As such it has a strong cursive aspect and is usually written in ample curves.
As Islam extended farther east, it converted the Persians, who took to use arabic script for their own language; and they contributed to arabic calligraphy the Taliq and Nastaliq styles. The later is extremely cursive, with exageratedly long horizontal strokes; one of its particularities is that vertical strokes lean to the right rather than the more common left, making Nastaliq writing particularly well flowing.
Finally the most commonly used script for everyday use is Riqa, simple and easy to write, its movements are small, without much amplitude. It is the one most commonly seen.
The traditional instrument of the Arabic calligrapher is the qalam, a pen made of dried reed; the ink is often in color, and chosen such that its intensity can vary greatly, so that the greater strokes of the compositions can be very dynamic in their effect.
Indeed, Arabic calligraphy hasn't fallen out of use as in the western world; the
Arabic script, cursive by nature unlike the latin alphabet, is used to write down a verse of the Quran, a Hadith, or simply a proverb, in a spectacular composition that is often undecipherable. The composition is often abstract, but sometimes the writing is shaped into an actual form such as that of an animal. One of the current masters of the genre is Hassan Massoudy.
Web references and examples :
http://www.arabiccalligraphy.com/resources.htm - examples of arabic scripts.
http://perso.wanadoo.fr/hassan.massoudy/ - Modern calligraphies.