Indo-European languages began as Proto-Indo-European (PIE) in the 4th Millenium BCE. The Kurgan Hypothesis states the origin of IE to be by the Black Sea in the North Caucasus region – formulated in the 1950s by Marija Gimbutas, this remains the most popular hypothesis amongst all linguists on the original location of PIE.
The Italic branch descended into several branches, the Latino-Faliscan subbranch (the only surviving one to date) giving way to the Romance languages. These consist of Italian, Spanish, Portuguese, French, Romanian, and various others, descend from varieties of Vulgar Latin spoken across the Roman Empire.
Germanic is a branch originating in Northern Europe: East Germanic is now extinct, consisted of Gothic (Crimean Gothic went extinct in 18th Century), the most divergent Germanic branch; North Germanic consists of languages descended from Old Norse, spoken by the Viking peoples from Scandinavia; West Germanic is the largest branch, consisting of Dutch, English, German, and many smaller languages in a dialect continuum across Northwest Europe (e.g. Low Saxon).
Celtic languages appear similar to Italic languages (which includes Romance) in their proto-forms, though diverge very differently. Only the Insular Celtic languages remain (moreso a geographical grouping than a linguistic one), Continental Celtic (incl. Gaulish) having gone extinct in the 1st Millenium CE. The Celtic languages alive today include Irish, Welsh, Manx and Scottish Gaelic. These languages exhibit a special morpho-phonological phenomenon known as consonant mutation, where the first consonant of words changes in certain situations (reflecting older phonological environments no longer present), e.g. Irish: "cat" /kat/ vs "mo chat" /mo xat/ vs "ár gcat" /ɔr gat/.
Slavic languages diversified most recently, around 1000 CE. They consist of East Slavic (e.g. Russian, Ukrainian, Rusyn), West Slavic (e.g. Polish, Kashubian, Sorbian), and South Slavic (e.g. Bulgarian, Macedonian, Serbo-Croatian).
Indo-Iranian languages diverged from the European branches very early in IE history. The Iranian languages (incl. Persian) diverged and left the Indo-Aryan languages in North Indic region, the most notable ancient language here being Sanskrit, language of the Veda. Vernaculars (called Prakrits) descended into various daughter laguages, including Hindi-Urdu and Bengali. Other Indo-Aryan languages include Assamese, the most easterly native Indo-European language today.
Extinct branches of IE include: the Anatolian languages (e.g. Hittite, Luwian, Lydian) spoken in Anatolia (modern-day Turkey) and the Aegean Sea islands nearby, though some linguists classify Indo-Hittite as a predecessor to IE; the Tocharian languages of China.