For all the details on the Roman magistracies and how they developped over time, please do read my post which /u/XenophonTheAthenian has kindly linked.

You're asking about the differences between the Roman and the American magistrates. There are many. For on thing, the Constitution is almost sacred to Americans. It is the absolute foundation of the American system of government. The Romans on the other hand had no written constitution. They had a system of government which was based entirely on tradition and precedents. So you see many changes to their 'constitution', where they decide to change or reïnterpret tradition, or simply create a new precedent. For all their love of tradition, the Romans were quite pragmatic.

Another thing is that the American system of government is firmly based in the Enlightenment idea of the seperation of powers. Montesquieu was clearly in the minds of the Founding Fathers. There is an executive branch, with a clearly seperated legislative branch and slightly less clearly seperated judicial branch (it is influenced by the executive branch and legislative branch via the appointment of Supreme Court judges). Montesquieu based his idea of the Trias Politica partly on Classical Greece and Rome. That does not mean however that Rome had this clear seperation of powers. There were some balancing mechanics in Roman government, but it was not by seperation of powers. Once appointed, the two consuls had pretty much all the power the kings of old had.

What limited consuls was the fact that they were in office for only a year, that they could be vetoed by their collegue or (later) tribunes of the plebs, and that they would become or remain (whatever be the case) part of the senatorial class together with all other former high magistrates. With these political and social limits in mind, consuls could propose legislation, lead armies, perform rituals, speak justice, address the assemblies. It was a president, vice-president, supreme court judge, speaker of the house, all-in-one.

Over time these things changed a bit. Many of the judicial matters were delegated to praetors (who, by the way, in theory could do almost everything consuls did). Provincial government was also delegated to praetors. Consuls focussed on legislation and leading the army. (It's a lot more glamourous than presiding over courts or managing a province.) As the Roman empire grew, all these magistrates could have their appointment extended for a year. We call these the "pro-"magistracies: proconsul (instead of a consul) and propraetor (instead of a praetor). For details see my other post linked above. A bit later still it became common to give out these promagistracies in multi-year appointments. Consuls and praetors would serve in Italy for a year, and would then get sent abroad as a promagistrate for a number of years. This is the time period the Rome series is set in.

To get back on track and actually address your question, consuls aren't really comparable to anything known in modern day. Senators are less problematic, but still don't have a real modern-day equivalent. Senators all were former consuls or praetors (or questors, but let's forget about them for now). So once someone had served high office in government, he automatically became a member of the Senate and would remain a member for life or until a census determined they were too poor (can't have poor people in the Senate, can we). The Senate (and thus senators) had no official role in Roman politics. It was an advisory body - one whose advice should be heeded, but without official political power. The Senate did however have the power to approve state finances. This gave them a lot of informal power. Anything which required money (which is anything, really) could be blocked by the Senate simply by refusing to fund it. If you must compare the Senate to something, it's probably closest to the English House of Lords. But don't get carried away with that comparison. In fact, it's probably best if you forget I mentioned it.

Tribunes then are an absolute disaster to explain. If you thought the consuls and praetors were complicated, wait till you have read this. With 'tribunes' you probably mean tribunes of the plebs. There were other tribunes, most notably military tribunes who had a role in leading the legions. The tribunes of the plebs were entirely different though. These ten tribunes were elected by the plebeians. Plebeians are all citizens who are not patricians, and patricians are those citizens who can trace their lineage back to the original Senate, during the time of the kings. Early on the Roman Republic, the patricians had way too much power compared to the plebeians. After lots of social struggles, the plebeians got more influence in politics. It was decided that at least one of the two consuls had to be a plebeian and the assembly of the plebeians (usually referred to as "the Plebs") got to elect 10 tribunes every year to represent plebeian interests in Roman politics. They could propose bills and conduct trials, and had the right to give assistance to citizens against magistrates. Tribune of the plebs was a sacred position, which ensured the safety of the tribunes (the Romans were quite religious). The most important power the tribunes got was the power of veto. A tribune could veto any decision by any magistrate, even a consul. Again, this is such a broad and powerful office that nothing modern compares to it.

Now it is often said that Montesquieu got his idea of seperation of powers from the Romans. It should be clear to you now that that is nonsense. What he got from the Romans is the idea of a balance of power, of different offices/assemblies of the state acting as a control on each other. He eventually put that in the mold of his Trias Politica, and that is what the U.S. system is based on. But really the only similarity between ancient Rome and modern America is that idea of a balance of power preventing any one part of government becoming too powerful; everything else was so different that it's sometimes even hard to imagine how the Romans handled things (see also my post on imperium).

This answer by thr user now known as /u/Polybios should be sufficient to outline the basics of the Roman constitution. Mind you, it changed drastically over time, in part because it was unwritten--much of this development is addressed in the answer, but by thr 1st Century, B.C. things got awfully complicated