1. Anatolian (extinct)
* Hittite
* Luwian
* Palaic
* Lycian
2. Tocharic (extinct)
* Tocharian A
* Tocharian B
3. Indo-Iranian
* Indo-Aryan
* Sanskrit
* Hindi
* Urdu
* Bengali
* Marathi
* Punjabi
* Gujarati
* Sinhalese
* Romani
* Iranian
* Persian
* Kurdish
* Pashto
* Ossetic
* Tajik
4. Greek
* Ancient Greek
* Modern Greek
5. Italic
* Romance
* Spanish
* Portuguese
* French
* Italian
* Romanian
* Catalan
* Latin (extinct)
* Italic (extinct)
* Oscan
* Umbrian
6. Celtic
* Insular Celtic
* Irish
* Scottish Gaelic
* Manx
* Welsh
* Cornish
* Breton
* Continental Celtic (extinct)
* Gaulish
* Celtiberian
7. Germanic
* North Germanic
* Swedish
* Norwegian
* Danish
* Icelandic
* Faroese
* West Germanic
* English
* German
* Dutch
* Afrikaans
* Yiddish
* East Germanic (extinct)
* Gothic
8. Balto-Slavic
* Baltic
* Lithuanian
* Latvian
* Slavic
* East Slavic
* Russian
* Ukrainian
* Belarusian
* West Slavic
* Polish
* Czech
* Slovak
* South Slavic
* Bulgarian
* Serbian
* Macedonian
* Slovene
9. Albanian
10. Armenian
This list includes the major branches of the Indo-European family. There are also numerous smaller branches and subgroups within these main groups. It's also important to note that the relationships between some of these branches are still debated among linguists.
Keep in mind that languages are constantly evolving, so the boundaries between branches are not always clear-cut. The classification of Indo-European languages is a dynamic and complex field.