Most languages on the planet are gregarious things; so they group together in large organisations, or families, of languages with which they have something in common... their relatives. English, for example, belongs to the Germanic branch of the Indo-European language family; whereas Mandarin belongs to the Chinese branch of the Sino-Tibetan1 language family. These families can be huge; for example, Indo-European languages are spoken by about half of the world's population.
On the other hand, some languages exist with no links to any other languages. These anti-social types, known as isolates, have grammatical, stylistic, and cultural features which no other language on earth has anything like. For instance, English and German - two Indo-European languages - are not isolates. They share a lot in common: the way they form words, their grammar, their style, and a lot of their vocabulary. They are also related to many other languages, from Danish to Sanskrit. True isolates, however, have no such similarities with any other language.
How did Linguistic Isolates Come About?
The short answer to this question is that no one knows. Most linguists just don't have a clue how isolates came into being. Many languages considered to be isolates today may one day be proved not to be. This is because languages, which have not been studied properly, or which are nearly extinct, may share links with others which have not yet been found.
Linguists believe this because of languages like Ket, spoken by a few thousand people in Siberia. This language was for many years thought to be an isolate but actually had a few relatives (Arin, Assan and Kott), which all died in the 18th Century, because people stopped speaking them. So Ket is an isolate in that it has no living relatives; but, in the true sense, it isn't an isolate because it once had a family.
Korean was also once thought to be an isolate; but most linguists now accept that it is part of the Altaic language family, and therefore very distantly related to languages like Turkish, Mongolian and Kazakh, now that it has been thoroughly studied.
Linguists believe that most isolates will one day be proved to have some as yet undiscovered links with other languages. They will turn out to have families after all.
Are There Any Real Isolates Then?
Yes, there are. Some are listed below
Basque is a language spoken by about 660,000 people in Northern Spain and South-West France. It is Europe's only living isolate.
Japanese is a favourite for linguists who are always trying to find links between Japanese and other languages; but they have never come anywhere close. Japanese is unrelated to any other language, despite having some borrowings2 from Chinese. There are 120 million speakers of Japanese.
Haida is a language of Alaska and British Columbia, with 350 speakers.
Hadza is spoken by 200 hunter-gatherers in Southern Tanzania.
The four languages above are among the better known isolates. There are plenty of others, mostly spoken by dwindling numbers of people in remote areas of the globe. This means that many will die out before they are ever studied or recorded properly; most will probably be extinct before their isolation can ever be determined as truthful or not.