Part 1: The First Age
Elrond was born towards the end of the First Age, at the Havens of Sirion. This was the dwelling place of the elves who had escaped the sack of Doriath, and the remnant of Gondolin who had survived the fall of the city in I.510. His mother was Elwing, Dior's daughter of Doriath, and his father was Eärendil, son of Tuor and Idril of Gondolin. He had one brother, Elros. One draft of the Tale of Years refers to them as twins.
When Elrond and his brother Elros were still young, the sons of Fëanor attacked the elves of Sirion, seeking to regain the precious Silmaril that Elwing had carried from the ruin of Doriath. Elrond and Elros were taken captive by Maglor, son of Fëanor, but he took pity on them, and cared for them for a while. It is not known exactly when Elrond and Elros parted from Maglor, but they probably did not remain with him for very long. At the end of the first age the Valar summoned the sons of Fëanor to return to Valinor, and Maglor reluctantly joined his brother Maedhros in rebelling against this summons and in seeking to fulfill the oath to regain the Silmarils.
After the attack on Sirion, Eärendil and Elwing managed to reach the shores of Valinor, and pleaded with the Valar to intervene on behalf of both elves and men in Middle-earth, for both peoples suffered under the shadow of Morgoth, the Dark Lord. Because they had been willing to enter into peril on behalf of the people of Middle-earth, and because they were of mixed elven and mortal descent, the Valar gave Eärendil and Elwing the choice of which people they would be counted among. This choice was also given to their sons, Elrond and Elros.
Elrond chose the kindred of the elves, and became a master of lore and wisdom. Elros chose mortality, and became the first king of Numenor, and the father of the line of kings that gave rise to Elendil and Isildur, and - eventually - to Aragorn.
Image: detail from "Elrond and Elros were taken captive" by Kasiopea. For full image and credits see the Artists' Impressions page.