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History of the Abacus:
Wikipedia and other sources speak of Somerians,Babylonians, Egyptians, Greeks and Romans using counting boards where rocks, boards or disks were slid around a board to keep up with the sum or reminder.
Salamis tablet (originally thought to be a gaming board), used by the Babylonians circa 300 B.C., discovered in 1846 on the island of Salamis.
The earliest known Chinese Abacus, sliding bead frame or suànpán is 200 BC or 1100 AD. These Chinese Abaci can do addition, subtraction, multiplication, division, square root and cube root.
China and surrounding countries still use an abacus in local business, especially where electricity is not reliable. Most Asian and European countries used and taught the use of the abacus up to 1900. Russia taught abacus in schools until the 1980's. The blind use a modifed version (beads tighter on rods).
Counting Board Competition: Hindu-Arabic numerals versus the counting board. Gregor Reisch 1508

Numbers are represented by wooden beads, systematically arranged in a frame. Compared to pen arithmetic or written math the abacus is far better. Time wise one could not write down the addition or subtraction numbers as fast as one could move the beads on an abacus even with a small amount of practice. There have been many contests of the abacus verses written math and verses the electronic calculator.
Chinese Abacus set to begin on 0
The Chinese Abacus consists of a rectangular wooden frame (average is 20 centimeter or 8 inch high) and is divided lengthwise into two unequal parts by a horizontal beam or bar with nine to 13 or more columns and 7 beads per column; two above the center bar and five below. The beads above the beam are called altobeads and those below it hypobeads. One altobead is equal to 5 hypobeads. The columns increase by 10 every time you move to the left.
Chinese Abacus set to the number 26.

It is critical to learn not only the techniques of bead math but also how to move the beads with which fingers so as to have the least chance of mistakes. Move the hypobeads with the thumb and index finger and the altobeads with the middle finger. After becoming an abacist it is like a typewriter or keyboard where you rarely look down at the beads except to see the final product.

There are 17 simple guides or secrets to memorize for addition and 17 similar ones for subtraction. To add the ability to do multiplication you only need to know the multiplication table up to 9 x 9 and the techniques. In division there are around 18 guides.

Sources: The Fundamental Operations in Bead Arithmetic, Wikipedia,