The Arabic word qibla means 'direction' of prayer, and this mosque is said to have two. One facing Mecca, and another above the entrance (look where everyone is staring). According to popular Muslim history, during a prayer here in AD 624, the Prophet received a revelation to change the direction of prayer from Jerusalem to Al Masjid Al Haram (the Grand Mosque), in Mecca, and so in mid-prayer he (or a local imam, depending on which narration you believe) immediately turned almost 180 degrees.
The significance of this event, seen as a defining moment for the religion, has elevated the status of this mosque, and it is now visited by pilgrims alongside other important sites in Medina. It's off limits to non-Muslims, as with all locations within central Medina.