The Indo-European Family

The term Indo-European was introduced in 1816 by Franz Bopp of Germany and referred to a family of languages in Europe and Asia (including Northern India, Iran, Afghanistan, Pakistan, and Bangladesh) that were found to have a remarkable structural relationship. It turns out that Sanskrit, Greek, Latin, Hittite, Old Irish, Gothic, Old Bulgarian, Old Prussian, and other languages share surprising attributes, meaning that most European languages and many of the languages of Iran, Afghanistan, Pakistan, Bangladesh, and India belong to the Indo-European family. 

The Indo-European family is divided into several branches or subfamilies. It consists of numerous Indo-Iranian languages, including Sanskrit, Hindi, and Farsi (Persian); Greek; Baltic languages such as Lithuanian and Latvian; Celtic languages such as Breton, Welsh, and Scottish and Irish Gaelic; Romance languages such as French, Spanish, Catalan, and Italian; Germanic languagessuch as German, English, and Swedish; and Slavic languages such as Polish and Serbian. 

This family has been the one most studied. It is also the one with the greatest number of surviving ancient documents and the one for which genetic links can be established with absolute certainty.

Here is the list of Indo-European languages presented in their respective subgroups:

Group

Subgroup

Languages

NOTE: † = Extinct language

Indo-Iranian

Indian
Sanskrit†, Hindi, Urdu, Bengali, Marathi, Bihari, Gujarati, Punjabi, Oriya, Rajasthani, Nepali, Assamese, Bundeli, Sindhi, Konkani, Pahari, Singhalese, Santali, Gypsy, etc.
Iranian
Avestan†, Persian (Farsi/Dari/Tajik), Afghan (Pashtu), Kurdish, Balouchi, Hazara, Aimak, Ossetian, Talyshe, Tat, etc.
Greek
Ancient Greek†, modern Greek
Italic or Romance languages
Oscan†, Umbrian†, Venetic†, Messapian†,  Raetian (Raetic)†
Latin† (mother language of Romance languages)
Italian, French, Spanish, Catalan, Portuguese, Galician, Mirandese, Provençal, Sardinian, Romanian, Romansch, Ladino, Friulian, Dalmatian†, Sicilian, etc.
Celtic
Gaulish†
Breton, Welsh, Cornish †
Irish, Scots, Manx†
Germanic
Gothic†
Danish, Swedish, Norwegian (Bokmål and Nynorsk), Icelandic, Faroese
English, Friesian, German, Dutch, Afrikaans
Baltic
Old Prussian†, Lithuanian, Latvian
Slavic
Polish, Czech, Slovak, Sorbian
Serbo-Croatian, Slovene, Bulgarian, Macedonian
Russian, Belorussian, Ukranian (and Ruthenian)
Armenian (isolate)
Armenian
Albanian (isolate)
Albanian (Tosk and Gheg)
Various isolates
Hittite†, Tocharish†, Lykian†, Lydian†, Luwian†, Phrygian†, Thracian†, etc.