Origins
The word 'counter' in English is actually several words that have converged into a single spelling, each with a distinct etymology. The most important are the noun 'counter' (a flat surface for business), the noun 'counter' (a disc or token for counting), and the prefix or adverb 'counter-' (against, in opposition). The first two share a root; the third does not.
The noun 'counter' as a flat surface comes from Anglo-Norman French 'countour,' from Medieval Latin 'computātōrium' (a place for computing, a counting room), from Latin 'computāre' (to count, to reckon, to compute). The Latin verb combines 'com-' (together) and 'putāre' (to reckon, to think, to prune, to settle accounts). A counter was originally a table used by merchants and money-changers for performing calculations. The flat surface was essential because merchants counted using physical tokens — small discs or coins placed on the table and moved to represent different values, much like an abacus.
These counting tokens were themselves called 'counters,' from the same root. The connection between the table and the tokens is direct: the counter (surface) is where you manipulate counters (tokens) to compute (count together) your accounts (reckonings). Board game pieces are still called counters in British English — a survival of the medieval counting disc.
Latin Roots
The root 'putāre' is remarkable for its range. In its oldest Latin sense, it meant 'to prune' — to cut back branches. This extended metaphorically to 'to settle, to reckon, to clear up' (pruning accounts as one prunes trees) and then to 'to think, to consider' (reckoning in the mind). From these senses, 'putāre' generated 'compute' (to reckon together), 'dispute' (to reckon apart, to disagree), 'reputation' (how one is reckoned by others), 'impute' (to reckon toward someone), 'deputy' (one reckoned as a substitute), and 'amputate' (to prune away, to cut off).
The prefix 'counter-' (against, in opposition) has a completely different etymology: it comes from Latin 'contrā' (against, opposite), through Old French 'contre.' When we say 'counterattack,' 'counterargument,' or 'counterclockwise,' we are using this Latin 'contrā' root, not the 'computāre' root. The convergence of these two etymologies into a single English spelling is a historical accident, not a semantic connection.
The German word 'Kontor' (an office, a trading post) comes from the same Medieval Latin 'computātōrium' and preserves the commercial meaning clearly: a Kontor was a trading house, especially of the Hanseatic League. The French 'comptoir' followed the same path and was used for colonial trading posts — the 'comptoirs' of French India, for instance.
Figurative Development
The evolution from counting table to shop counter to kitchen counter follows a clear trajectory of metaphorical extension. Once the flat surface was no longer used for counting but for displaying and selling goods, 'counter' became the generic term for any flat horizontal surface over which business is conducted. The kitchen counter, which arrived in American English in the twentieth century, represents the final step: a flat surface for domestic work, with no trace of the original counting function.
'Computer' is the ultimate descendant of the same root. First attested in 1613, 'computer' originally meant a person who computes — a human calculator. The electronic computer, named by analogy in the 1940s, is etymologically a machine that does what the counter (table) and the counter (disc) did together: reckoning numbers, computing sums.