As you suspect, the answer to this question is extremely complex. There were a huge variety of different status classifications for the urban centers of the Roman Empire, and each province was different than the next. In simple terms, there were provincial capitals, coloniae, municipia, "civitates / poleis", and vici, all of which had different situations. Among the colonia, there were regional differences depending on the province, and some provinces had very few if any coloniae. For the municipia, there were as many constitution types as there were cities, and endless kaleidoscope of privileges from and responsibilities to the Roman state. The civitates were "self-governed" typically, and in whatever way they wanted or had done it in the past, again dependent on region. An urban center's status was treasured by its inhabitants and was viciously defended, and typically the ins and outs of its precise existence within the Roman system was hashed out in negotiation and concession, often over centuries.

The provincial "capitals" had no special status, except that the provincial governor tended to make his home base there, and they were sometimes the site of larger concilia for the surrounding regions, where representatives from multiple urban centers would convene. Depending on the province, its capital could be a colonia or a lowly vicus. So Mogontiacum (Mainz) was the capital of Germania Superior and an important legionary base, but was only a vicus until the Late Empire. By contrast, a town like Emerita Augusta (Mérida) in Hispania Lusitania was a very large and important colonia.

Having said all that, there are some basic similarities. Virtually all coloniae and municipia, regardless of where they were, had tripartite governments modeled more or less on those of old Italian towns. The central magistrates of a colonia were the duoviri (or duumviri), the "two men," who were (usually) elected from among a narrow pool of oligarchic local candidates. With them were locally-elected aediles, slightly more junior in prestige, and together they formed a quattuorviri. The duoviri were technically judicial officials, but also were responsible for summoning the other organs of government to order. The aediles were the "mayors" and ran everything, usually via subordinates, including the upkeep and new construction contracts for public monuments and spaces, temples, sewers, roads, aqueducts, baths, etc. The aediles also oversaw the economic life of the city, including market functions, coinage, food supply, port regulation, taxes, etc. Sometimes there might be quaestores to handle the fiscal duties, also elected. All these men were drawn from a narrow pool of candidates based on local wealth and prestige, and were Roman citizens. Working with them was a second branch, the "council", an advisory body of varying size also drawn from local big-shots and usually called the ordo decuriones. They deliberated on problems and guided the town's business in concert with the quattorviri (or not). They often were the body that communicated with the regional or Imperial capital and its officials, who regulated citizenship, who ran the local festivals and temple personnel, etc. Nota bene that the term "decurion" has several meanings in Roman matters. Here, it just means "councilors of local town." The third organ was the assembly, the gaggle of Roman citizens at large in the local community, who theoretically elected the magistrates and passed the decrees and proposals put to them by the magistrates and the council.

The governments of the municipia were much the same, again with differences large and small depending on what town, and where. Typically, not all the members of a municipium had Roman citizenship, again depending on the place. But the quattuorviri and any other elected officials would have citizenship, or would acquire it upon rising to office. This was an important mechanism for ensuring local elites "bought in" to the Roman system, and it worked very well. See Greg Woolf's Becoming Roman (Cambridge 2009) for a very thorough exploration of this topic in early Imperial Gaul.

Major cities which were not coloniae or municipia were often termed civitates. Like the above, there was a gigantic variety of different arrangements for these, but the overarching principle is that they were not Roman citizens. Usually they were under Rome's thumb in some way or another (usually multiple), and paid in to the system via taxes, tribute, and obligations. They often kept their own local governing styles, but sometimes also adopted some sort of system like that outlined above. It's too complicated to dig into here, but some were immunis (almost completely free of obligations) or libera (free from certain obligations).

All that above is on the Western side of things. In the East, which was predominately Greek, the terms were different, if not the substance. Most of the Greek cities of the East were under some kind of tripartite system, with magistrates called "archons" instead of duoviri, a boule instead of a council, and an ekklesia assembly. Instead of aediles in inscriptions, you might find astynomoi doing roughly the same job. In general, the older, more storied Greek poleis tended to have more titles and magistrates than the Western veteran colonial cities, many of which sprung up out of the ground like mushrooms after Actium, but the overall result was the same. Some Greek urban centers were civitates and were left to do their own thing, as long as they money kept flowing. Others became coloniae, like Corinth.