The first city that your starting Settler unit has constructed becomes your Capital (with a free Palace), until you decide to build a new Palace in another city, which causes that city to replace your existing Capital, and your Palace to be moved to that city.
A city which contains the Forbidden Palace National Wonder or the Versailles Great Wonder will become a capital for the purposes of maintenance. The city banner will show the following icon: . Cities nearby these "secondary" capitals will have a reduced maintenance penalty from distance. Secondary capitals do not benefit from the increased Commerce or Production from Bureaucracy.