At its core, the Senate is best compared to the House of Lords or the Canadian Senate rather than the modern American Senate. Much like the House of Lords, they served a dual role as both a member of the legislative and judicial branch. Let me explain.

To become a senator, the only real qualification was that you had been an elected magistrate at the level of quaestor of above. After that, there were varying and rather fickle qualifications that were determined by the censor, who could (and did) expel senators from the rolls for 'ethical' reasons (hence the term censorship).

As I mentioned this was more comparable to the House of Lords (pre 2005) than the American Senate. If you aren't familiar with it, the House of Lords (again, pre-2005) acted as both the upper house of the legislature of the United Kingdom, as well as one of the courts of final appeal. Thus is operated in a dual role (much in the same way that the Prime Minister is both head of the legislative and de facto head of the executive branch in the form of his role as head of cabinet). This is because the Senate, for most of the history of Rome, also controlled the law courts (if, indirectly).

The actual function of Roman democracy is a bit different than you might imagine. The senate was not technically one of the assemblies. There were three main assemblies: the comitia centuriata, based on the property divisions for soldiers which elected the highest ranking magistrates and could declare war, the comitia tributa, based on the tribes of citizens, which elected lower magistrates, and the concilium plebis, which was the Plebian's main assembly. By the mid to late Republic, the later two were the most important for passing laws (for example, it was the concilium plebis that the Gracchi brothers used to pass their legislation).

So what is the Senate's role in all this? Well, the Senate were the ones who could decide who could stand for elections in the comitia centuriata, and these positions were the dominant aspects of the executive branch. The senators also decided who would be appointed to proconsular and propraetor positions (governorships) of the provinces, and indeed, exercised almost exclusive control over the running of those provinces day-to-day. Finally, the senate had to approve any outlays from the treasury, a power they used often to thwart populist legislation. For example, those same Gracchi got around that control by directly assuming control over a large donation from the dead king of Pergamum 'to the Roman people', which they did not interpret as 'to the Roman treasury'.

Thus, the best way to think of the Senate is to break up their de jure and de factor roles. De jure, the role of the Senate was to pick who could stand for election and advise the assemblies, with said advice supposed to be taken very very seriously. De facto, this role as 'advisors' meant that they controlled the courts and the most lucrative sources of income, as well as who could actually stand for election. When you combine this with the fact that the senate also contained the richest men in the city, you can see how they might use their power. Ultimately though, their power legally speaking was based more on tradition than fact, and individual wealthy and ambitious men of the senatorial class demonstrated that the power of the assemblies, when used in a direct and and focused manner, could overcome the 'advisory' role of the Senate. The unfortunate result was that the Senate, lacking any legal measures, took extralegal steps to assert it's authority starting with the Gracchi and continuing with events like Cicero's actions vis a vis the Cataline Conspiracy. In short, the inability of the Roman system to deal with the imbalance of power between the senate, magistrates, and assemblies (i.e. a lack of effective checks and balances) was one of the key reasons for it's eventual collapse.

The Senate was not like a modern elective legislative body--technically it didn't even have legislative authority, but in practice its advice was usually heeded. More importantly wasn't a based on election, rather it was an assembly of statesmen. If you are elected to an office of the Republic you automatically are inducted into the Senatorial Order, and after that you remain a member unless you kicked out for whatever reason.

All very answerable questions, but gonna ask you to specify in a few ways - also check the FAQ as well, there should be something in there.

Do you mean specifically the Empire or the Republic? Do you mean the Senate exclusively or the Assemblies as well?