The exclusive surviving member of the "Balkan" clade is Albanian, a linguistic isolate among Indo-European languages. Three different Paleo-Balkan languages (Illyrian, Thracian, and Dacian) have each been proposed as the ancestor of modern Albanian, hence these four at least might be argued to constitute a "Balkan" family. Exactly how these and other old Balkan languages including Macedonian and perhaps Paionian might be related to Albanian, to each other, and to the Indo-European language family more generally is a much debated topic supported by too little objective evidence (i.e., ancient texts). Yet because these and certain other ancient languages of the Balkans are generally if not provably considered to be Indo-European and somehow related to one another, and because they were geographically clustered, they are grouped for convenient reference.
Albanian's Indo-European ancestry was first suggested in the 1850s, but was not fully established until the early 20th century: Albanian appears exotic against other Indo-European languages because it has suffered great lexical change from the reconstructed parent. The broken history of the Balkans sheds light on the near-total absence of PIE-inherited words and their replacement with loans from Greek, Latin, Slavic, and Turkish over the course of twenty-five centuries. The surviving PIE-inherited words, meanwhile, balance considerable phonological change with striking conservatism, such as the preservation of all three PIE tectal series before front vowels.
We encounter Albanian already much weathered in the earliest preserved texts. Among all major Indo-European families, Albanian has the latest first attestation: the language is first recorded in various 15th century AD marginalia. A mid-16th century missal-book signals the real beginnings of a literature. (N.B. A recently reported discovery could change this picture, but -- until the subject document is scientifically dated, published, and thoroughly analyzed -- the report cannot be considered as more than a rumor. We must ignore rumors, as forgeries of "ancient" documents are not uncommon.)
What is collectively called Albanian actually comprises two dialects, Gheg and Tosk, thought to have split from a common Albanian stock some 1500 years ago. Respecting their geography, the River Shkumbin serves as a rough latitudinal divide with Gheg to the north (widely spoken in Serbia and Montenegro, less in Albania and Macedonia) and Tosk to the south (primarily spoken in Albania, but also by diaspora populations in Greece and Turkey). The Orthodox Christian literature of the 1600s helped to establish Gheg as the Albanian written standard for three and a half centuries, until the stark reversal of this situation in the aftermath of World War II when Tosk-speaking Communists consolidated their power over the region. Gheg speakers were displaced and marginalized, and so remain; Tosk went on to become Albania's language of government, media, education, and literature.