For the answer to your question, I turn to History of Pittsburgh by Sarah Killikelly, who did historians a great favor by studying the evolution -- if that's the word -- of city government, so the rest of us don't have to.
According to Killikelly, the first attempt to establish a local government in Pittsburgh dates to 1792, when there was an abortive effort to establish the city as a township. Nothing appears to have come of it, except for a pair of guys being named "supervisors of roads, etc." (Come to think of it, I wonder if these guys are still available: City government today could use someone supervising roads -- not to mention the etc.)
But by 1794, the city's growth was large enough that we needed some government to retard it. And so, the city elders petitioned the state to let the city become a borough. This was done, according to the state Act, so that "nuisances, encroachments of all sorts, contentions, annoyances and inconveniences ... should be prevented, and for promoting rule, order and good government in the said town."
Or not. The original borough of Pittsburgh was small, about the size of Downtown today. The first census taken in 1800 showed a population of just under 1,600 people. Yet the city government, establishing a precedent that would last for centuries, already seemed large. It had two "Chief Burgesses" and four "Assistant Burgesses," not to mention a High Constable, Town Clerk, two assessors, two supervisors and three borough surveyors/regulators. Two of these officials, an assistant burgess and a supervisor, resigned after two full days in office. Were they tired of the political bickering? Had they fulfilled all the items on their political platform? Or were they simply trying to cash in on their government pension? Hard to say. But this might well be the first and last time in Pittsburgh history that an elected official got out while the getting was good.
Despite this impressive infrastructure, voters were unhappy with government services, establishing another political tradition. In 1804, the borough government was changed by the legislature, which held that the current government was "insufficient to promote convenience, good order and public utility." The number of burgesses was cut from one to two, with the council expanded to 13 members.
According to the new charter, Pittsburgh was to be "forever a borough." That would make a great title for a made-for-TV movie about Dormont, if someone ever made one, but it wasn't much of a plan for Pittsburgh. As Killikelly puts it, "There seems to have been no consciousness at the time that this second borough charter would, in a little more than a decade, be judged inadequate." As trade and manufacturing expanded, Pittsburgh outgrew its government again, and so Pittsburgh finally became incorporated as a real, bona fide city on March 18, 1816. Apparently, they didn't even need a pro sports team to earn the designation.
Pittsburgh's original city government included a mayor and not one but two city councils, known as the Select and Common Councils. They were sort of like the Senate and House of Representatives, except without any of the actual dignity. Under the terms of the first city charter, the governor appointed the city's recorder and a panel of aldermen, who functioned as city magistrates. The mayor had all sorts of power; along with the city's alderman, he could try defendants for many crimes. Nowadays, he's more likely to be the subject of grand-jury investigations himself.
By Pittsburgh standards, the original city government proved a remarkably stable structure; it wasn't until nearly a century later that the two branches of city council were merged into a single body. (Councilors have tried to compensate for the loss of the second council by becoming twice as argumentative, so be sure to thank them for maintaining the same level of constituent service.) The mayor's power steadily increased, rising to its mid-century apex under David Lawrence and eventually reaching a Götterdí¤mmerung conclusion under Tom Murphy. On the bright side, the city has managed to return to the redundancy that has defined it since the outset: We now have not one but two financial oversight boards, at least one of which appears to serve no real purpose except making trouble for our lame-duck mayor.
Which raises an important question: Is it too late to take another shot at becoming a township?