Appearing as far back as 2700BC, the abacus is one of the first calculating devices ever created (preceded by a few computation tablets). Despite its antiquity, I had an abacus at my house growing up. I must admit I never had the need to use it, nor was I a fan of arithmetic. With practice though, it was on par with modern electronic calculators. Surprisingly, abacuses are still used today. The device doesn't require any electricity or writing supplement, and is useful for fast calculation when neither of those are present.
An abacus, at its core, holds different tokens at varying heights/widths within a two dimensional frame. Dependent on the type of abacus, the tokens will either slide horizontally or vertically on rods inside the frame (although the rods/slidable tokens allow for faster operating, the calculations of an abacus could just as easily be done with rocks being moved on the ground).
Users manipulate the alignments of those tokens in order to perform arithmetic. The tokens/beads are arranged to form a number, then an operation is performed (like addition or subtraction) and the resulting form of the beads can be read to be the result of the operation. Since the result is the representation of a number in the abacuses, subsequent operations can be performed to the result without any need to rearrange the board further.
The representation of a number in abacus form, as well as the operation depends on the type of abacus. The Japanese abacus, a soroban, has its rods oriented horizontally, and the Russian abacus, a schoty, has its rods oriented vertically. Typically, each rod represents a different digit or unit depending on the base. Both the orientation of the rods and the number base system used greatly changed the way in which an abacus was represented; An abacus working with Roman numerals differed from those in base 60 (like a Sumerian abacus), which different from our modern base 10 Arabic number system.
Bi-quinary system
The soroban used a bi-quinary system to encode base-10 digits into the rods. Each rod would be a digit, where the digit itself was represented by the positioning of the beads. The rod would be split into a top half and bottom half. The top half would have two positions (bi portion), and the bottom half would have four (quinary portion). The digits would be represented by the following encoding:
The first portion either represents a 5 or 0, and the second portion of the beads represents 1 through 4 (adding the two portions would be the digit the rod represents). The top would have one bead, and the bottom 4. The number would be represented by how many beads you would move up/engage. So to represent the number 7, the upper bead for 5 is engaged, and two of the lower beads are pushed up. For the number 4, the upper bead remains unengaged, and all four lower beads are pushed up.
The source of the above images is the Wikipedia article for soroban, which I highly recommend: https://en.wikipedia.org/wiki/Soroban.