The language families of Europe fall into a few broad categories. There are the Indo-European languages, which include the Romance, Germanic, Slavic and Celtic subgroups, along with Greek and Albanian. The Iranian languages and most of the languages of India are also Indo-European. Then there are the languages of Finland and Hungary, which are hypothesized to be of a broader Finno-Ugric family.
Whatever the validity of this cluster, the relationship of Hungarian and Finnish to languages which are extant deep into Eurasia, beyond the Urals and into Siberia, are not disputed. Turkic and Semitic families have a toehold in Europe via Turkish and Maltese. And finally, you have the Basque dialects. Basque is not related to any other language in the world; it is a linguistic isolate. There have been attempts to connect Basque to languages in the Caucasus, but these are highly speculative conjectures.
So where did Basque come from? A common assumption is that Basque is the autochthonous speech of the Iberian peninsula, perhaps related to the pre-Latin dialects extant to the south and east of the peninsula (the Romans arrived on the scene at a time when Spain was also partially dominated by Celtic tribes). Many go further and assert that the Basques are the pure descendants of the first modern humans to arrive on the European continent, heirs of the Cro-Magnons. Even if this claim is a bit much, many would cede that the Basque populations derive from the hunter-gatherers who were extant on the continent when the Neolithic farmers arrived from the Middle East, and Indo-European speakers pushed in from the east.
