It plays only a little part - Patrician is mainly trading simulator, and s bit of RTS (naval battles).
In case you want a political career (city mayor), you need high reputation in your town. Reputation is achieved mainly by supplying the city with enough resources, donating to the church and other charities, creating jobs (by building factories), and so on.
There is a bit of political maneuvering once you become councilor. You might bribe once or twice a year your fellow councilors, and they would vote with you on council meetings. You might need to fulfill stock requests from the prince, and sometimes give him 'gifts' to keep him satisfied and not attacking the town.
Same things apply if you want to raise your political career to the top of the Hanse - you raise your reputation by doing other cities favors, bribing important people in other cities, and so on.
And if you are not interested in political career, you can concentrate trade and become very rich. Or you can concentrate on fighting pirates, or you can become a pirate. Overall, political maneuvering plays only a small part in Patrician.
HTH