I have been having trouble finding actual information regarding this. I have heard of Rome being a Republic at one point? It is always referred to as an Empire ruled by the Caesars? Also, a lot of people seem to believe it to be the penultimate Empire that only collapsed because it just got old. So, I would like to have some real information and details to go off of. Was it really all that stable? WAS it ever a Republic? Did it have an Imperial structure, and if so, did it overtake the Republic or was the Republic made part of it?
I apologize in advance if this is a difficult one to answer, but finding answers on my own has been a pain.
That is an extremely broad and wide question. Honestly, if you want to understand this, you're better served reading some introductory books or articles on ancient Rome. There are some nice ones on our book list.
There is also our FAQ on the Roman republic and the Roman Empire.
I particularly recommend this thread giving a broad level overview of the Roman Republic (written by me, but consisting largely of links to more in-depth discussions by people who know more than I do)
But honestly, given your current state of confusion, I expect that thread is likely to only confuse you more because it all assumes a certain level of background information. So to help you get started, here's a very, very high-level overview:
The Roman Kingdom
According to the Romans themselves, Rome was founded in 753 B.C. and began life as a petty kingdom, a city state ruled by a king who was advised by a council of elders called the Senate. Early on in their history they became dominated by their older and more powerful neighbours, the Etruscans, and the Roman kings became more tyrannical until the last of them, Tarquinius Superbus, was overthrown by the heroic Brutus and Rome became a republic.
How much of this is actually true is up for debate. ( as per u/Tiako in the linked thread.) Scholars agree that the first kings of Rome are purely mythological and Rome actuallys started later. Most think the later kings did exist. Either way, we know that Rome was a republic somewhere from the 5th century B.C. onwards.
The Roman Republic
So yes, ancient Rome was a republic. But it was not "the Roman empire" at this stage: it was one city state among many in Italy, waging war against its neighbours. Over time, it came to dominate its surrounding region, and through fighting the Etruscans, Gauls in the north of Italy, and Greeks in the south of Italy, it got control of all of Italy. From there on it got in conflict with the Carthaginians (See: Hannibal) from North Africa, and then the various Greek kingdoms in the eastern mediterranean. By conquering all these lands Rome became an empire in the modern sense of the word: a state that rules over a vast geographical area and dominates many different peoples. But politically, it was still a republic.
What does it mean to be "a republic" in this time? That is a very complicated question, actually. Again, I refer you to the link I gave above that gives an overview of the various terms and institutions. But in very brief: Rome held elections each year. Through these elections, it chose magistrates that governed the city, performed various public functions, and also served as generals in the Roman army. (Early in the republic, the highest elected officials (consuls and praetors) led the armies. Later, when Rome conquered more distant lands, it would send magistrates abroad to serve as governors and generals in the provinces, but only after their normal term of office had expired. So they'd first serve in Rome as consul or praetor, and then go off to a province where they'd lead an army and govern in the name of Rome.)
All free Roman male citizens could vote in these elections, but Rome was not a democracy like ancient Athens was. The voting system was quite rigged to favour the elites. Still, elections were important and the Roman people could and did make their will felt. Roman citizens could also vote on laws directly, though they could not propose laws themselves. (A magistrate would bring a law before them and they could vote yes or no.) Again, the system was actually extremely complicated and the links above can tell you more.
In addition there was the Senate, a term you've very likely heard before. The Senate was not elected and senators served for life (generally...) and mostly consisted for former elected magistrates. It had in theory and advisory function, but in practice their immense prestige gave them a lot of power.
This system worked well enough until the 1st century BC, when a series of civil wars brought the whole system down. The why and how is too complex to go into here, but there were repeated periods of anarchy where military strongmen seized control of the state and led armies against one another. Caesar was one of these, as were men like Pompey, Marius and Sulla. Anthony of Anthony-and-Cleopatra fame was also one of them.
The last of these was Gaius Octavius, who later changed his name to Augustus, and whom we remember as the first Roman emperor. (He was also Julius Caesar's adopted son and took his name, hence "the Caesars.")
The Roman empire: the Principate
So, in the conventional understanding Augustus restored order by seizing direct control over the state and the army: he became emperor, and Rome became an empire ruled by a single man, the emperor.
It is -of course- more complex than that. Augustus himself claimed he was just a citizen trying to serve the state, who had done much to save and restore the republic. He called himself "first among equals" or "princeps," hence the name "the principate." This was propaganda, but it was not completely untrue. Rome still held elections, even if the emperor could sometimes "advise" who should win. The senate remained and senators continued to be the most important people in the empire, often serving as generals in the armies and governors in the provinces. Except now the emperor had the final say.
The short of it is that the early empire was a system of informal checks and balances, where the emperors held most of the power but did still share it with old republican institutions.
This is the period that is what most people think of as "The Roman Empire" and it featured many of the things we know best about it. It was the time of the professional legions, when the Colliseum was built. It is when all those colourful characters like Nero and Caligula lived.
The Roman empire: The Dominate
This system lasted until the 3rd century A.D., when a series of crisis threatened to (and for a time did) rip the empire apart. Foreign invasisions, plague, a fiscal and economic crisis, and incessant civil wars wrecked the empire. At one point, some 30 emperors ruled in a period of 50 years, and many more would-be emperors attempted coups or revolts that did not succeed. The FAQ has more information on this if you're interested.
Eventually, a series of soldier-emperors from the Balkans, called the "Barracks emperors", managed to restore order and re-conquer bits of the empire that had split away. The system that emerged from this is called the Dominate, and was quite different from what came before. More details can be found in this thread by u/mp96
In brief: this was the Roman empire without those left-over Republican trappings. A more centralised state, with a bigger bureaucracy, and an emperor who now was without a doubt the lord and master over all he surveyed. Hence "dominate", from "dominus" or lord. Think the kind of emperor where you kneel and kiss his robe rather than where he sits in the senate with his "fellow senators" from time to time.
This centralised system functioned quite well and Rome saw another period of prosperity and power, but one interesting side effect was that it could only realy function when the emperor was nearby, precisely because everything was so centralised. They solved this by having multiple emperors at once, so a crisis in two places would not cripple the system. This eventually led to a split in an Eastern and a Western Roman empire, though this split was never formal and never intended to be permanent, and several times the empire was re-united under a single ruler.
The Byzantine empire
Typical school history has it that the Roman empire fell in 478, when the last Western Roman emperor was disposed. The fall of the Roman empire is another huge debate. (here is an old thread I wrote on it, very long so feel free to skip
But either way, the eastern half of the Roman empire went on much as the Dominate had done. The Arab conquests would eventually take over much of the former Roman empire in the east, but the Romans themselves hung on in Greece and Turkey until well into the middle ages. (1204 or 1453 AD depending on how you count it.)
Most modern histories call this medieval eastern Roman empire the "Byzantine empire" and its history could (and does) fill another small library worth of books.
In conclusion
So was Rome stable? The main takeaway from this post could be: