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. 

map of Europe and Asia

The Indo-European family is divided into several branches or subfamilies. It consists of numerous Indo-Iranian languages, including Sanskrit, Hindi, and Farsi (Persian); GreekBaltic 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 languages such 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.

The Indo-European Family

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

Group

Subgroup

Languages

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