In technical, economic terms, a monopoly is a business that has complete control over a given market. That is, it is the only business that sells a given good or service in a given market. We sometimes use the term in a less technical way. We can use it to refer to a business that is extremely big and controls almost all of the market. For example, in the time period that you are asking about, people referred to Standard Oil as a monopoly even though it sold “only” 90% of the refined oil in the United States at that time.
For the most part, Americans disliked the idea of monopolies. This was not true of all Americans, of course. There were some people who believed that monopolies were a natural result of competition and that the winners of the competition were the best companies around. Even so, most Americans felt that monopolies were harmful. They felt that monopolies used their power over the market to charge prices that were too high and to abuse consumers and workers in other ways. It was for this reason that there were attempts at creating antitrust laws (laws limiting or banning monopolies) as early as the 1880s.