Step 1: Understanding the Concept:
The Mughal Dynasty ruled the Indian subcontinent for several centuries. The question asks to identify the person who established this dynasty.
Step 2: Detailed Explanation:
The Mughal dynasty was founded by Zahir-ud-din Muhammad Babur in 1526. He was a descendant of Timur on his father's side and Genghis Khan on his mother's side. Babur invaded India and defeated Ibrahim Lodi, the last Sultan of Delhi, at the First Battle of Panipat in 1526, thereby laying the foundation of the Mughal Empire in India. The other rulers mentioned—Akbar, Shah Jahan, and Aurangzeb—were his descendants and subsequent powerful emperors of the dynasty.
Step 3: Final Answer:
The founder of the Mughal dynasty was Babur. Therefore, option (A) is the correct answer.