All-pairs testing or pairwise testing is a combinatorial software testing method that, for each pair of input parameters to a system (typically, a software algorithm) tests all possible discrete combinations of those parameters. Using carefully chosen test vectors, this can be done much faster than an exhaustive search of all combinations of all parameters, by "parallelizing" the tests of parameter pairs. The number of tests is typically O(nm), where n and m are the number of possibilities for each of the two parameters with the most choices.
The reasoning behind all-pairs testing is this: the simplest bugs in a program are generally triggered by a single input parameter. The next simplest category of bugs consists of those dependent on interactions between pairs of parameters, which can be caught with all-pairs testing. Bugs involving interactions between three or more parameters are progressively less common, whilst at the same time being progressively more expensive to find by exhaustive testing, which has as its limit the exhaustive testing of all possible inputs.
Many testing methods regard all-pairs testing of a system or subsystem as a reasonable cost-benefit compromise between often computationally infeasible higher-order combinatorial testing methods, and less exhaustive methods which fail to exercise all possible pairs of parameters. Because no testing technique can find all bugs, all-pairs testing is typically used together with other quality assurance techniques such as unit testing, symbolic execution, fuzz testing, and code review.
Also see my other posts on effective test case generation techniques.