Short answers, pulled from the story.
Over 3.4 billion people, about 42 percent of the global population, speak an Indo-European language as a first language. This is by far the most of any language family. Ethnologue counts about 446 living Indo-European languages, of which 313 belong to the Indo-Iranian branch.
Proto-Indo-European is the reconstructed common ancestor of all Indo-European languages, spoken sometime during the Neolithic or early Bronze Age around 3300 BC. It left no written records and has been rebuilt entirely from its descendant languages. It was an inflected language with a complex system of 15 stop consonants.
The academic consensus supports the Kurgan hypothesis, which places the Proto-Indo-European homeland on the Pontic-Caspian steppe in what is now Ukraine and Southern Russia. It is associated with the Yamnaya culture and related archaeological cultures during the 4th and early 3rd millennia BC.
Sir William Jones lectured to the Asiatic Society of Bengal in 1786, arguing that Latin, Greek, and Sanskrit had sprung from a common source that perhaps no longer exists. Earlier observers included Thomas Stephens in 1583 and Filippo Sassetti in 1585, but their notes did not lead to further inquiry. Marcus Zuerius van Boxhorn proposed a shared ancestor he called Scythian in 1647.
Hittite is the earliest recorded Indo-European language. The Anitta text, written in Hittite and dated to 1700 BC, is the oldest known text in any Indo-European language. Even older are isolated Hittite and Luwian words found in Old Assyrian texts from the 20th and 19th centuries BC.
The Indo-European family has ten major branches: Albanian, Anatolian, Armenian, Balto-Slavic, Celtic, Germanic, Hellenic, Indo-Iranian, Italic, and Tocharian. Anatolian and Tocharian are extinct, while the others contain living languages. All descend from Proto-Indo-European.
The centum-satem division splits Indo-European based on how the word for hundred developed, and it was named by Peter von Bradke in 1890. In satem languages, including Balto-Slavic and Indo-Iranian, the palatovelar sounds became sibilants. In centum languages such as Latin they merged into plain velars, giving the hard k of centum.