Zetetic has its etymology in the Greek ζητείν, "to seek." Elench is: "that part of an argument on which conclusiveness depends; that which convinces or refutes an antagonist; a refutation"
The Zetetic Elench are described as a group of explorers. They splintered off from the Culture proper as they felt that it had become too unchanging. They seek out new experiences and perspectives and hope to be changed by these. Despite its somewhat alien sound, their name is actually made up of two fairly obscure English words: zetetic elench might reasonably be read as "empirical refutation", which is a fair description of the group's modus operandi - refute the Culture's values by their own experiences. An alternative rendering might be "the search that convinces" or "the proving search".
Ships of the Zetetic Elench are grouped into fleets, which all travel in the same small area of the galaxy, but operate independently most of the time. Contact within a fleet is normally just a regular checking in with a trusted outside contact, but if one of their number go missing a coordinated search will begin.
Although no longer part of the Culture, relations between ships of the Culture and the Zetetic Elench are friendly and closer than that which exists between Culture ships and ships of the other Involveds.