What 2 branches of government did Rome have?

The Roman Republic did not have branches of government as we understand them today. The Roman Senate's powers were not very clearly defined, nor was it an elected body. Basically is was a fairly regular meeting of representatives of the most important and noble Roman families who - certainly in their own mind - in a way represented 'the State'. Your family could only be 'elevated' to Senatorial rank, either by being elected as Quaestor (during the Republic) or later, by appointment of The Emperor. Executive power was firstly held by two annually elected consuls (who also proposed new laws and were the State's supreme judges) who were at the same time generals of the army for that year. In all, there were five executive ranks, and after the consuls:

1. Praetors, who executed part of the consuls' duties. In peacetime those were mostly religious; during the many wars, they usually held military commands under the supreme commandership of the consuls.
2. Censors, who were responsible for various censuses and were the 'overseeers of public morals'- although that had practically nothing to do with 'sex' at the time. Through the census they kept the State up to date on its population, its wealth and of the state of essential services. Although they were less powerfulful than praetors, the jobs were usually held by former consuls and the office had great prestige because of that.
3. Aediles, two pairs of them each year, two plebean and two patrician. Like the questorship to be mentioned under 4, this was a job held by younger members of the important families who were trying to climb the slippery pole to consulship. Aediles were responsible for maintenance of public buildings and the organizing or Rome's festivals. This last responsibility was a major one: it could show your organizational skills, show how much money you could and would spend on them and if done well, it could put you very much in the public eye for later and higher office.
4. Quaestors. who were responsible for overseeing the public finances and those of the armies. Strangely enough, this was considered the most 'junior' of the public offices.

All these office holders (all elected for 1 year, except for the censors who were elected for 18 months) acted largely independently from each other. A check only on the Consuls' activities was instituted in the form of the "Tribunes of the plebs" who could veto decisions and laws made by the Consuls. The office of Tribune had been instituted as a check on what the Patrician clans and families did in office: this was because most of the above offices could only be held by Patricians. A Tribune on the other hand could only be someone from a Plebeian family. Tribunes could also act as 'military Tribunes" and then hold lower command. It was a powerful office, not only because the Consuls' act could be blocked but also because a Tribune could independently propose laws and have them approved in self-convened public meetings 'as long as these affected only the Plebeians'.

Finally, in times of crisis a consul or former consul could be appointed as Dictator, mostly for a period of 6 months. He then could take all the decisions normally taken by any of the other office holders and he could appoint people in an office instead of having them elected.