The numbers Americans and Europeans use most commonly, having the form 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Although referred to as "Arabic," they originated among the Hindus in India and were transmitted to Arabia via a translation of the Indian Siddhanta, which was brought to Baghdad in A.D. 772.
In the Arabic world, numbers were originally written out in words and, eventually, certain Arabic letters began to be used as numbers (much as the Greeks had done). Once the Siddhanta had been translated into Arabic, the Hindu numeral system found therein was adopted by Arabic merchants, while the original letter-based system was retained by Arabic astronomers.
After a split in the Arabic empire, the Hindu number system eventually mutated into two different ones; one remained in the eastern portion of the empire, and closely resembled numerals still in use in the Middle East today. The other system eventually found its way to Toledo, in Spain, populated by dispossessed Arabs. When Toledo was toppled in 1085 by Christian mercenaries, a tremendous amount of Arab knowledge was revealed to the West, although some European scholars, who had studied in Spain, were already using and teaching the new numerals.
The first Arabic numeral proponent in Europe was a man named Gerbert (who later became Pope Sylvester II). John of Seville, in the twelfth century, translated the works of the Arabic mathematician Al-Khwarizmi, and as a result the Arabic numerals became well-known, and were originally described by the word algorism, a corruption of Al-Khwarizmi's name which eventually mutated into the modern word algorithm. (The word algebra, by the way, also has a similar origin, coming from the name of another Arabic mathematician, Al-Jabr"].) In addition to the numerals, the concept of the zero (0), unknown in Roman numerals, was introduced to Europe.