Broadly speaking, there were two distinct scripts in the early centuries of Islam: cursive script and Kufic script. For everyday purposes a cursive script was employed: typical examples are to be seen in the Arabic papyri from Egypt. Rapidly executed, the script does not appear to have been subject to formal and rigorous rules, and not all the surviving examples are the work of professional scribes. Kufic script, however, seems to have been developed for religious and official purposes.
The term Kufic means "the script of Kufah," an Islamic city founded in Mesopotamia (modern day Iraq) in AD 638, but the actual connection between the city and the script is not clear. Kufic is a more or less square and angular script characterized by its heavy, bold, and lapidary style. Its letters are generally thick, squat, and unslanted, and it was particularly suitable for writing on stone or metal, for painting or carving inscriptions on the walls of mosques, and for lettering on coins. Professional copyists employed a particular form of Kufic for reproducing the earliest copies of the Quran that have survived. These are written on parchment and date from the 8th to the 10th century. They are mostly of an oblong as opposed to codex format. The writing is frequently large, especially in the early examples, so that there may be as few as three lines to a single page. The script can hardly be described as stiff and angular; rather, the pace is majestic and measured. With the high development of Arabic calligraphy, Kufic writing became an exceptionally beautiful script. From it, there were derived a number of other styles, chiefly medieval, in North and Central Africa, Spain, and northern Arabia.
Kufic went out of general use about the 11th century, although it continued to be used as a decorative element contrasting with those scripts that superseded it. About AD 1000 a new script was established and came to be used for copying the Quran. This is the so-called naskhi script, which has remained perhaps the most popular script in the Arab world. It is a cursive script based on certain laws governing the proportions between the letters. The two names associated with its development are Ibn Muqlah and Ibn al-Bawwab, both of whom lived and worked in Mesopotamia. Naskhi was always employed chiefly for writing on papyrus. In time, it evolved into innumerable styles and varieties, including the ta'liq, the riqa', the diwani, and the thuluth, and became the parent of the modern Arabic writing.
Distinctive scripts were developed in particular regions. In Spain the maghribi ("western") script was evolved and became the standard script for Qurans in North Africa. Derived ultimately from Kufic, it is characterized by the exaggerated extension of horizontal elements and of the final open curves below the middle register.
Both Persia and Turkey made important contributions to calligraphy. In these countries the Arabic script was adopted for the vernacular. The Persian scribes invented the ta'liq script in the 13th century. The term ta'liq means "suspension" and aptly describes the tendency of each word to drop down from its preceding one. At the close of the same century, a famous calligrapher, Mir 'Ali of Tabriz, evolved nasta'liq, which, according to its name, is a combination of naskhi and ta'liq. Like ta'liq, this is a fluid and elegant script, and both were popularly used for copying Persian literary works.
A characteristic script developed in Ottoman Turkey was that used in the chancellery and known as divani. This script is highly mannered and rather difficult to read. Peculiar to Turkish calligraphy is the tugra (tughra), a kind of royal cipher based on the names and titles of the reigning sultan and worked into a very intricate and beautiful design. A distinctive tugra was created for each sultan and affixed to imperial decrees by a skilled calligrapher, the neshani.
There has always existed in the Islamic world a keen appreciation of fine handwriting, and, from the 16th century, it became a practice to assemble in albums specimens of penmanship. Many of these assembled in Turkey, Persia, and India are preserved in museums and libraries. Calligraphy, too, has given rise to quite a considerable literature such as manuals for professional scribes employed in chancelleries.
Source: Encyclopedia Britannica