Short answers, pulled from the story.
Kapilavastu served as the capital for the Shakya clan, a political entity known as a gañasañgha or republic. King Śuddhodana ruled over this territory alongside Queen Māyā during the late Iron Age.
Prince Siddartha Gautama lived within the walls of Kapilavastu until he departed at age twenty-nine. Early Buddhist texts describe it as the childhood home of Gautama Buddha.
Chinese monks Faxian and Xuanzang traveled to the region during the medieval period and recorded accounts describing Kapilavastu as part of Central India. Their travelogues became crucial for later scholars searching for the lost city.
Samuel Beal translated the records of Hiuen Tsiang in 1884. His publications included Si-Yu-Ki and The Life of Hiuen-Tsiang which guided archaeologists toward specific areas near the border.
Two modern locations claim to be the historical capital today with Tilaurakot sitting within Nepal while Piprahwa lies in Uttar Pradesh, India. Both sites contain archaeological ruins that suggest ancient settlement activity dating back to the 5th century BCE.