Cities in the roman empire had a large degree of autonomy. They mostly used their own codes of law and administered themselves independent from the roman governor of the province (of course they had obligations like paying taxes or other fiscal liabilities). Over the years though many cities adjusted their constitutions after an more roman/latin model.

All roman cities shared one defining attribute: A well defined group of local aristocrats. The cities were governed by these nobles (who had a very localised powerbase, they were not part of the imperial aristocracy) who were all part of the "Ordo Decurionum", a kind of city council (how you got into office of course differs from town to town, could be hereditary, could be elected). On average, there were about 100 members of the Ordo Decurionum in every city.

The ordo decurionum took care of almost all public matters of the city (defense and foreign policy of course were handled by roman officials). For example, they organised religious festivities, races, took care of the town infrastructure, local law enforcement. The members of the Ordo Decurionum served as judges (if there were no different local structures). Roman governors also used to regularly delegate their own duties to the cities to minimize their own administrative efforts. This led to extremely high financial burdens and often led to the impoverishment of cities who lacked the income necessary to pay for all of this.

Other offices in the cities: The italian cities and roman/latin colonies mostly copied the roman urban administration. These cities had their own aediles (who took care of policing, and the markets), quaestores (who served in many parts of the administration), praetores (mostly judges) and (as their equivalent of the consules) "Duoviri" (literally the two men, some cities used "iure dicundo" as an addition) who were basically the majors of the town. In some cities there were 4 of those, then they were called quattuorviri. Every 5 years the "duoviri quinquennales" (which means the two men who get elected every 5 years, romans were pretty clear in their office naming ;D) who had the duty to conduct a census, make a list of all members of the ordo decurionum and to controll the citizens lists. To get into on of these offices (and thus into the ordo decurionum) you needed to pay the city some money, the "summa honoria". The amount needed ranged from about 800 (parts of north africa) to 38000 (carthargo) sestertii. All of these officials needed the approval of the Ordo Decurionum of their city to do anything. Much like the senate in republican times, the Ordo Decurionum was the heart of municipal politics.

Also, as in rome city, the priestal offices were extremely prestigious. Examples are the Augustales and flamines, who oversaw the imperial cult, or the augures, who watched the birds.

Many cities also still had their own people's assemblies, like the boule of alexandria or the ekklesia of ephesos. These had varying amounts of power of course. They were more common in italia and the italian colonies and of course the greek cities of the eastern mediterranean.

One thing to add: While there were formal offices, the administration of the city was regarded as common task for every member of the local nobility. Even if they had no office at the moment, the members of the Ordo Decurionum had to spend big parts of their own money to aid the city. This of course led to the co-option of everybody who was rich into the city council, just because the city needed the money. The Ordo Decurionum was a sure path for the newly rich to get a first taste of politics. Thus, the Ordo Decurionum was the number one tool for climbing up the social ladder if you were for example the first succesful merchant of you family.

Sources: Mostly Karl Christ's "Geschichte der römischen Kaiserzeit" (History of the roman Empire)