Germanic (Teutonic)-Western : German (High German spoken in Central Europe, Low German closer to the coast), Yiddish (spoken by East European Ashkenazi Jews), English, Dutch, Afrikaans (South Africa) Frisian. Northern: Norwegian, Swedish, Danish, Icelandic. Eastern: Gothic (extinct).

Romance -French, Spanish, Portuguese, Italian, Romanian, and some minor languages: Provencal (in southern France), Catalan, Sardinian, Rhaeto-Romance, Ladino (spoken by Sephardic Jews).

Celtic-Formerly spoken over a wide territory; now a few million speakers of Gaelic (Irish), Scots-Gaelic, Welsh (Cymrag), Breton (in Brittany, France).  The ancient Gauls were Celts.  Many place names in central Europe are of Celtic origin.

Slavic-Three branches. East:  Russian, Byelorussian, Ukrainian. West:  Polish, Sorbian (a tiny group in Germany) Czech, Slovak.  South:  Serbian, Croatian, Slovene, Bulgarian, Macedonian.     

Baltic-Lithuanian, Latvian; the closest relatives of the Slavic group; in contrast to most other modern Indo-European languages, the Baltic languages have changed very little since ancient times.

Albanian-Two dialects, Gheg and Tosk; contains many Turkish loan words.

Greek-Modern Greek has changed greatly from ancient Greek.

Armenian- Spoken by several million people north of Turkey and Iran.

Hittite-One of several extinct IE languages formerly spoken in what is now Turkey.

Tocharian-two extinct IE languages formerly spoken in western China.

Iranian-Dozens of languages spoken by millions in Iran and Afganistan; include Farsi (modern Persian), Pashto, Kurdish.  Ancestors called themselves Aryans, meaning "strong" or "noble." (This ancient word appears in the modern toponym Iran.)

IE languages of India-Dozens of languages descended from ancient Sanscrit; include Hindi, Urdu, Bengali, Punjabi; now spoken by over a billion people. 

Non Indo-European languages of Europe and vicinity include: 

Basque- The sole surviving relic of western Europe's pre-Indo-European past.

Etruscan- An extinct non IE language of Italy.

Finnish and Estonian- belong to the Finno-Ugric language group of northeastern Europe; the IE peoples didn't get to this part of Europe.

Hungarian-Distantly related to Finnish; spoken by descendants of North Asian invaders from the ninth century AD. Several non-IE peoples invaded Europe after the fall of Rome, but only the Hungarians stayed and preserved their language.

Georgian-a language isolate spoken by a few million people north of Turkey.

Turkish, Arabic, and Hebrew are important non Indo-European of Southwest Asia.

Number of speakers of the world's major languages

Today, roughly half the world's population speak an Indo-European language. Most of the world's major languages are IE  (except: Chinese, Arabic, Japanese, and Malay-Indonesian).

                                        Number of speakers (in millions as of 1994)                                                               Number of speakers

                                                Total          Natives only                                                                             Total          Natives only

Mandarin                               930         827                                         Portuguese                           179         167        

English                                  463         319                                         Malay-Indonesian                152           49

Hindi                                      400         327                                         Japanese                              126         124        

Spanish                 371         326                                         French                                   124           72

Russian                                291         172                                         German                 120           98

Arabic                                     214         182                                         Urdu                                         98           95

Bengali                  192         184                                         Punjabi                    92           90