Kairouan (about 800 AD)
The first buildings that were built in the Islamic Empire were designed by Greek architects who had already been living in the area when the Arabs conquered it. Because of that, these buildings look a lot like earlier buildings in the area - Late Roman Empire buildings. But because they were now building Islamic mosques and not Christian churches, these Greek architects were able to experiment with some new forms, developing a new Islamic style. One of the earliest mosques is the Dome of the Rock in Jerusalem, from the 600s AD. It's octagonal, like Hadrian's Pantheon, instead of being cross-shaped like a Christian church. In the late 700s AD, the new Arab rulers of North Africa marked their new territory by building great mosques like the one at Kairouan (modern Tunisia) and the one at Cordoba in Spain.
In the Abbasid period, beginning about 800 AD, the capital of the Islamic empire moved further east, to Baghdad, and so the caliphs needed a lot of new beautiful palaces and mosques built in Baghdad. Because Baghdad was in the old Sassanian Empire, the architects who lived there followed Sassanian architectural traditions, and these buildings, like the mosque at Samarra, looked very different from the ones built by the Greek architects.
In the end, though, the Islamic Empire made it so easy to travel around that all the architects got to know each other's styles, and there got to be one main style of building all across the Islamic Empire. As the empire broke down into a lot of smaller kingdoms, the ruler of each kingdom needed to show how important he was, so he built mosques and palaces in his own capital. The Fatimids, for example, built the Al-Azhar mosque in Cairo in the 900s AD. In Spain in the late 1200s AD, the Almohads, built their own palace at Granada, the Alhambra.
The Ottoman sultan built the last great Islamic building before 1500 AD - his palace in Istanbul, which he built in the late 1400s AD.