Almost all languages in Europe, Persia and Northern India derive from a single language called Proto-Indo-European. Here is a simplified family tree:

The tree shows how languages derived from one another. 

 English is closely related to Dutch and German.

Through a similar process the Sanskrit spoken in North India changed into the modern languages of of the region: Hindi, Urdu, Punjabi, Bengali and others.

Ancient Persian evolved into Farsi, Kurdish and Pashto.

Each language family is a group of related languages with a common ancestor. 

Languages in the same branch are sister languages that diverged within the last 1000 to 2000 years (e.g. Latin gave rise to the romance languages in the Indo-European family).

Languages in different branches of the same family are called cousin languages. For most families these languages would have diverged more than 2000 years ago.

Languages in the same family, share many common grammatical features and many of the key words (called cognates), especially older words, show their common origin. Here is the word month in several Indo-European languages:

The difference between a language and a dialect can be political rather than linguistic. 

For example, Croatian and Serbian are linguistically closely related dialects of the same language. However, they are written in different scripts (Latin and Cyrillic) and are spoken by people of different religions living in Catholic Croatia and Orthodox Serbia respectively. They are called different languages for political reasons.

Macedonian is considered by Bulgarians as a dialect of their language while Macedonians themselves consider it a separate language. Since Bulgaria has long claimed Macedonia as part of its territory, the reasons for each view are obvious!

Low German (spoken in Northern Germany) and Dutch (Netherlands) are linguistically dialects but politically separate languages. Low German and Swiss German are mutually unintelligible but are both considered to be German. There are more differences between Italian spoken in different cities in Italy than between Danish, Norwegian and Swedish.

We look at ancient forms of languages - what kinds of plants, animals, tools, and landscapes their speakers were talking about, and try to figure out where the source language originated, how it dispersed and changed. 

We can make inferences about where people traveled and how they made their living. Linguistic analysis can be used as a tool for dating developments such as plant domestication or the invention of the chariot. 

This kind of study is the field of linguistics, anthropological linguistics, and geography of language. 

Indo European Language Family

Most current theories put the origin of Indo European languages in eastern Anatolia. 
Others place it further north on the Russian Steppe:

There are four dominant theories about how Indoeuropean languages came to spread out from their source areas and dominate Europe and Western Asia.  it was spread by 

1) bronze-age warriors expanding out into Europe and SW Asia some 4000-5000 years ago (Childe 1950, Gimbutas 1980). 

2) traders who used it as a lingua franca along trade routes throughout western Eurasia - especially after the domestication of the horse on the Ukrainian steppes some 5000- 6000 years ago (Outhoudt 2002).

3) early agriculturalists - the first to domesticate wheat, barley, cows and sheep around 7000 to 8000 years ago (Renfrew 1987, 1992). 

4) hunter gathers during a period of rapid warming after the end of the ice age when they quickly repopulated Europe and western Asia after the end of the last ice age (Younger Dryas) around 9,000-10,000 years ago (Adams 2002). 

The radiocarbon 14 dates fluctuate too much and the archaeological evidence is spotty. We may never know whether Indo European was spread by hunters, farmers or soldiers. 

What Languages spoken in Europe are not part of the Indo European language family?

Basque - is  spoken in the Pyrenees between France and Spain. There are no other languages in its language family. The language counts in 20s, a property borrowed by French numbers above 60. Recent research may indicate a link with Etruscan, a pre-Latin language from Italy.

The Uralic Language family include languages that originated in the Ural mountains in Russia. In the Finnic Branch, Finnish and Estonian are closely related. Languages in the Ugric Branch (like Hungarian) are very different having separated from the Finnic ones around 3000 years ago. Hungarian's closest relatives (Ostyak, Vogul) are found in central Siberia.

The majority of the languages in this family are spoken in Siberia (Mordvin, Komi, Nenets) apart from Sámi which is spoken in Lapland (northern Scandinavia).

The Altaic Family is named after the Alti Mountains, in Central Asia. These people were nomadic horsemen living in the plains. One group migrated towards Europe, the other group migrated towards the Korean Peninsula and the islands of Japan

Turkish is the most westerly member of this family as well as the most spoken. Many of the others are spoken in former USSR republics (Azeri in Azerbaijan), Turkmen (in Turkmenistan), Kazakh (in Kazakhstan), Kirghiz (in Kyrghystan), Uzbec (in Uzbekistan, land of Genghis Khan), Uigur (in Western China east of the Pamir Mountains).

Mongolian is found in Mongolia (where it is written in the Cyrillic script) and Northern China (with a script that goes down rather than horizontal). Korean and Japanese are the most easterly Altaic languages.

The Caucasian Family is named after the Caucasus Mountains,  between the Black Sea and the Caspian Sea. Georgian and Chechen are the main languages. They are known for their large number of consonants. This area has a tremendous number of different languages. See Krysstal's external link