It was dominated by the wealthy so by Greek standards it was more of an oligarchy. But Polybius in the 2nd century B.C. has a hard time pigeonholing the Roman system and calls it a mixed constitution because it was a combination of oligarchical, democratic, and monarchical elements. The people voted for the magistrates (who became members of the Senate) but the voting in the Centuriate Assembly (comprising 193 "centuries") was heavily weighted in favor of the wealthiest citizens, so the election was usually decided before the poorer citizens got to vote. (Once a candidate had a majority of the possible 193 votes, the voting stopped, just like how the World Series ends once one team has won 4 games).
Certainly it was true that during the Republic a small number of noble families dominated the consulship--it was rare for a "new man" with no consular ancestors to be elected.
Exactly, what elections there were became more open to the hoi-polloi *after* the so-called republic (which was run by an unelected oligarchy) was dumped, and there was an Emperor.
Ironically, Rome became an empire when it started conquering other cities, starting with Ostia, but probably wouldn't have swallowed the Med basin and western and Balkan Europe had it not been for the attacks *on* Rome by the Gauls and then the Carthaginians. They had to improve their military skills or perish.