Xenoglossy is the ability to speak a language without having learned it. This sounds like a really neat trick if you can manage it.
However, a typical place to find this rare word is the Journal of Parapsychology. That’s because the ability is regarded as a psychic phenomenon. It might come about because a person has been regressed to a previous incarnation through hypnosis. Or a medium might be in communication with a spirit person who speaks another language. The OED dates its first appearance to 1914; it’s from Greek xenos, stranger or foreigner, plus glossa, language; another spelling is xenoglossia.
It sounds as if it’s related to speaking in tongues, which is regarded among Christian groups such as the Pentecostal and Charismatic movements as evidence of the presence of the Holy Spirit (though the phenomenon is found in many religions and has been recorded from the earliest historical times); the formal term for that is glossolalia (the second half from Greek lalia, speech).
But from a language point of view the difference is profound: in xenoglossy the implication is that a real language is being spoken that is intelligible to native speakers and in which the person can converse, while glossolalia is a succession of meaningless syllables interpretable only through faith.