How do I choose a mayor?
Eventually you will want to have three mayors. The level of the mayor doesn't count, only the stats he has -- in fact, it's better to recruit a low-level hero with a high stat than a high-level hero. This is because the hero levels fast if he's level 1 (takes less experience) than if he's level 30. So if you recruit him when he's level 1 and he has 70 politics, by the time he's level 30 he'll have 99 politics, whereas the guy you recruit at level 30 with 70 politics will have only gotten to maybe level 35 or 38, and only have like 76 politics. So build up your feasting hall to level 3, and keep your eye on the Inn and start gathering heroes such that you have one with high politics, one with high intelligence, and one with high attack. The other stats don't matter, only the one they're being chosen for. Get your politics mayor first, then your intelligence mayor, and then your attack mayor. You won't need the attack mayor for some time.
POLITICS: This is the mayor you will want in office almost all the time. High politics decreases construction time, which means that your buildings will finish faster, and the fortifications on the walls (traps, ATs, etc.) will finish faster as well. The main thing, though, is that high politics gives you a significant increase on your hourly resource production rates. You get an addition % for each point of politics -- so a mayor with 70 politics will increase your hourly yield by 70%. Really significant!
INTELLIGENCE: Intelligence in a mayor will decrease the time it takes to research a tech. It's a fairly significant difference, so the longer your techs start taking, the more valuable this will be. Luckily for us (and our production bonus with the politics mayor) we only need to have the intelligence mayor in office when you START a research project. So, to give an example, I have my politics mayor in office by default. I see that my last research has finished, so I go to my feasting hall, click "appoint mayor," and switch from my politics mayor to my intelligence mayor. Then I go to my academy, click "research" on the tech of my choice. Once the research starts, I go back to my feasting hall and put the politics mayor back in. The research project will have the same quicker finish time that it had when your research mayor was in office.
ATTACK: Attack mayors are like intelligence mayors, but for troop training. The higher the attack, the quicker the troops will be created in your barracks. Like the intelligence mayor, the attack mayor only needs to be in office when you queue the troops. So for example, I decide I want to make 500 archers. I go to my feasting hall, click "appoint mayor," and select my attack mayor from the list. Then I go to my barracks, and decide I want to build them in batches of 100, so I can have the first 100 etc. sooner than waiting for all five to finish. So, I queue all 5 batches of 100 archers. Then I go back to my feasting hall and replace the attack mayor with my politics mayor. All 5 batches will still have the faster training time you had under the attack mayor.