Short answers, pulled from the story.
Mughal architecture is the style of building developed across the Mughal Empire during the 16th, 17th, and 18th centuries throughout the Indian subcontinent. It is characterized by large bulbous domes, slender corner minarets, massive halls, large vaulted gateways, and delicate stonework ornamentation. The style synthesized Indo-Islamic, Persian, Central Asian Timurid, and indigenous Hindu architectural traditions.
The Taj Mahal was commissioned by Mughal Emperor Shah Jahan in memory of his wife Mumtaz Mahal. Construction ran from 1632 to 1653, taking 22 years and requiring 22,000 laborers and 1,000 elephants at a cost of 32 million rupees. The chief architect was Ustad Ahmad Lahori.
Akbar's greatest architectural achievement was the walled capital city of Fatehpur Sikri, built near Agra starting in 1569 and completed in 1574. The city included the Jama Masjid, the Tomb of Salim Chishti, and the Buland Darwaza (Gate of Magnificence), which stands roughly 54 metres high and was commissioned in 1576 to commemorate Akbar's victory over Gujarat and the Deccan.
The Badshahi Mosque in Lahore, Pakistan, was commissioned by the Mughal Emperor Aurangzeb and constructed between 1673 and 1674. It is the largest Mughal mosque ever built and the last of the imperial congregational mosques in red sandstone. Its courtyard covers 276,000 square feet and can accommodate one hundred thousand worshippers.
Mughal architecture influenced the Indo-Saracenic, Rajput, and Sikh architectural traditions within the subcontinent. Beyond South Asia, one scholar documented similarities between Mughal designs and Ethiopian imperial construction projects under Susenyos I in the early 17th century, attributed to Jesuit connections between the two regions. Several mosques in Malaysia and the Omar Ali Saifuddien Mosque in Brunei also incorporate Mughal architectural elements.
Red sandstone and white marble were the signature materials of Mughal architecture, replacing the brick that dominated earlier Indo-Islamic building. White marble was initially used as cladding to finish sandstone structures, as seen in Humayun's Tomb, before being used to cover entire buildings such as the Taj Mahal. Under Aurangzeb, the later phase of the tradition saw a return to brick or rubble finished with stucco.