Most political terms are approximately 2 double moons long, with the only exception being the Respected Citizen, which is only one double moon. Upon registering your candidacy for an office with the mundane burgess, it is notated in your legend that you are running. After having sufficient clout to enter office, speak to the mundane burgess again and register once again for that office. He'll notice the sway you have and allow you to step in to office and record it in your legend.
While in office, try to do your best with your position. Remember why you entered office, to serve the city. Make the most of your political term, because it is short. Respected Citizens should try to set examples, Guards and Captains should patrol and be available to prevent crime. Demagogues should set up laws along with the Burgess. The Judge should be available to schedule and carry out hearings. You earned the clout to step into office, don't let it go to waste needlessly. While I say this, don't be overbearing as well. Know the line. ^_^
Citizens of a city can access a list of officials currently in term, called the "Guards List", which should be posted on several of the community slates around your city. This makes it easy for citizens to find an official if they need one.
Lastly, a word of congratulations. If you have stepped into office, you earned it. The road is a hard one, and often less traveled by Aislings after seeing the clout needed for office. Use your unique skills and talents to make the most out of your office and make your city prosper!