What is a language family, and what is a language isolate?
A language family is a group of languages descended from a common ancestor. A language isolate is a language with no demonstrable relatives at all.
How families are established. Not by similarity — borrowed words make unrelated languages look alike. The method is the comparative method, which identifies regular sound correspondences across vocabulary that languages rarely borrow: numerals, body parts, kinship terms, basic verbs.
If one language consistently has p where another has f in the same positions across many such words, that systematic pattern indicates common descent rather than contact. Grimm's Law describes exactly this between Germanic and other Indo-European languages, which is why English has father where Latin has pater.
The major families:
Indo-European — the largest by speakers, covering most of Europe, Iran and northern India. English, Hindi, Russian, Spanish, Persian and Greek are all relatives.
Sino-Tibetan, including Chinese languages and Burmese.
Niger-Congo, the largest by number of languages, including the Bantu languages.
Afroasiatic — Arabic, Hebrew, Amharic, Hausa.
Austronesian, spread remarkably across Madagascar, Indonesia, the Philippines and the Pacific.
Isolates are languages that cannot be placed in any family.
Basque is the most famous, spoken in northern Spain and southwestern France. It is unrelated to Indo-European and to everything else, and it appears to predate the Indo-European arrival in Europe — making it a survivor of a linguistic landscape otherwise entirely replaced.
Korean is often classified as an isolate, though relationships have been proposed. Ainu, Burushaski and several Amazonian and Papuan languages are others.
Why isolates exist. Either the relatives died out, or the relationship is too ancient to demonstrate. The comparative method has a practical time depth — beyond roughly 8,000 to 10,000 years, regular correspondences become indistinguishable from chance, so deep relationships may exist and be unprovable.
Sign languages have their own families, unrelated to the spoken languages around them.