First, someone has to propose the civic - you have to be a non-peon in your civ in order to do so. There'll be a bright green "propose civics" button, which leads to a menu to choose the civic you want to propose. After someone proposes a civic, you'll get an option to vote yes or no. It'll show you the current vote, allow you to change your vote as often as you want, and announce when anyone else in your civ registers a vote.
The vote might dissapear right away if the political minister vetos it. It doesn't currently seem to give the rest of the civ a chance to overwhelmingly support the proposal before processing the veto.