The Roman government (in its entire history from founding to fall) was a strange mix of a democracy and a republic. An interesting fact ist that the people of Rome took many of their ideas of government from the Ancient Greeks.
The Roman state was described as the republic (respublica) and its consuls, or chief magistrates, continued to be appointed even after the establishment of one-man rule under the empire, but in its pure form it lasted only until the beginning of the first century B.C.
At the creation of the republic, supreme power probably resided with a popular assembly, but early on the Senate became very influential, and the traditional formula, which survived for centuries, was S.P.Q.R. - Senatus Populusque Romanus - the Roman Senate and People acting together.
Since the Ancient Romans did not want one man to make all of the laws, they decided to balance the power of the government between three branches, there was first the executive branch, then the legislative branch, and finally the judicial branch.