IEEE C37.09 is the test standard accompanying the C37 series of high-voltage circuit breakers in North America. It describes how the characteristics defined by C37.04 are verified: interrupting capability, operating sequences, timing and behaviour under short-circuit currents.
Its role is the same as that of IEC 62271-100 internationally, and test sets normally ship with pre-configured test plans for both. The practical differences lie in the naming of the sequences, in some tolerance criteria and in how results are documented.
In installations with equipment of mixed origin it is common to find both references in the same report, and it is worth having an analyzer that lets you choose the standard against which each test is assessed rather than imposing one.