I particularly enjoy questions about our past that lead to an understanding of how we arrived where we are today. By understanding how Rome came to dominant the Mediterranean world, we understand a crucial step in the formation of Western culture, giving us a greater understanding of why things turned out the way they did. Countless aspects of European culture were learned from the Romans. Before the Roman Empire, much of Europe was populated by barbarian tribes such as the Celts, illiterate and often nomadic. By 400 AD the Romans had civilized much of Southern and Western Europe, introducing Roman government, alphabet, numerals, language, architecture, philosophy, law, roads, agricultural techniques, and ultimately Christianity. Later governments, particularly the US, would adopt many Roman ideas in their constitution. The world today would look drastically different had a small village on the River Tiber named Rome not grown to dominate the Mediterranean world.
Rome began around 750 BC as a collection of rural villages along the River Tiber. The local geography provided a number advantages: the Tiber provided navigable access to the Mediterranean, nearby salt flats were a source of wealth, the surroundings had excellent soil that until ~1000BC had been fertilized by numerous active volcanoes in the area, and the city was located on five hills, making it easier to defend.
I believe Rome also benefited from its relative open-mindedness in terms of other cultures and ideas. For example, Rome frequently found itself in need of a larger labor force, and throughout most of its history foreigners were offered citizenship if they moved to Rome. Earlier in its history, it was also not uncommon for foreigners to gain power, for instance around 600 BC Tarquin, son of Greek royalty, became king (Rome had 3 political phases: Kingdom (750-509BC), Republic (509-49BC), and Empire (49BC-476AD), and is said to have brought Greek influence to Rome. And, learning from the example of Alexander the Great, once a town surrendered to Roman forces, local nobility frequently retained positions of power in the town. In addition, captured peoples were welcomed as citizens of Rome and foreign religions were tolerated, as long as proper deference was paid to the Roman pantheon. With such tolerance, captured peoples were less likely to revolt, and Rome, open to new ideas, learned immensely from its neighbors, especially the Greeks.
Romans were at times enamored with Greek culture, and Roman culture was in many ways shaped by Greek culture. Cicero writes "It was indeed no little rivulet that flowed from Greece into our city, but a mightly river of culture and learning." Roman legend claims that Rome was founded by the twins Romulus and Remus, later claimed to be descendants of Aeneas, a Trojan prince who escaped Troy with a small group of settlers at the time Troy fell to the Greeks (perhaps around 1200 BC). While this is surely myth, it reflects Rome's great debt to Greek culture. In the beginning of its history, Rome was flanked by the Greek-influenced Etruscans to the north, and Greek colonies throughout Southern Italy and Sicily. Many of the areas that became part of the Roman empire had previously been part of Alexander's empire and had been heavily influenced by Greek thought. Later in its history, many Romans learned Greek or would even study abroad in Athens. Many of the ideas that made Rome capable of building a vast empire were learned from the Greeks: the alphabet, the use of numerals, law, philosophy, and technology. And while Rome was never purely a democracy as Athens had been, the plebians (common citizens) usually had a powerful voice in government through elected tribunes and direct votes, and could not easily be ignored by the non-elected branches of Roman government.
I also wonder to what extent Homer's epics, the Iliad and the Odyssey, shaped Roman history. His epics, perhaps written around 800BC, had an enormous influence on Greek and Roman thought and perhaps even history. The characters in these epics primarily seek glory. They are fiercely proud, intolerant of the slightest insult, and hold that lineage and bravery were more important than the pursuit of wealth. Alexander the Great (300's BC) was greatly inspired by Achilles, and some believe that having Achilles as a role model potentially drove Alexander in his endless pursuit of territory, in order to attain immortal glory. Similarly, I wonder if this pursuit of glory over wealth was behind the fantastic ambitions of some of Rome's great expansionist leaders.
Another aspect of Roman tradition that drove its expansionism was this: one of the most effective ways of climbing the political ladder was conquest of new territory. For instance, Caesar's conquest of Gaul ultimately lent him the support needed to become the first emperor in 49BC.
Also quite important was the culture of self-sacrifice for the good of Rome. Two dramatic examples of this occurred during the battle to quell a revolution of Rome's Latin neighbors in 341BC. The battle was led by the two "Consuls" (co-leaders of the republic, with only one year terms), Titus Manlius and Publius Decius Mus. Titus sent a squadron of cavalry, with his son Titus among the leaders, to do some reconnaissance but with strict orders not to take part in any fighting. His son, passing close to the enemy lines, was taunted by an enemy commander. Against his father's instructions, he agreed to a jousting contest with the commander and killed him, as the remaining soldiers looked on. When Titus discovered that his son had disobeyed him, he said "Titus Manlius, you have respected neither Consular authority nor your father's dignity. I believe that you yourself, if you have any drop of my blood in you, would agree that the military discipline you undermined by your error must be restored by your punishment. Go, lictor, bind him to the stake." The army watched horrified as Titus' son's neck was severed on his father's orders. The other consul, Decius, made the ultimate sacrifice. After ascertaining from a priest that victory could be assured if only he would "devote" himself (i.e. commit suicide in order to persuade the gods to take the lives of your enemy as well), he said a prayer, leapt on a horse, and rode directly into the enemy's ranks where he was promptly killed. These stories give you an idea of the extent of the devotion of (some of) the ruling class to Rome's cause. Such a culture of self-sacrifice surely played a part in Rome's success.
So, to sum up, much of the success of the Roman empire was due to its inheritance of Greek traditions, strategic location, interest in expansion, tolerance of and interest in other cultures, and a relatively well-organized governing class that valued, at least earlier in the empire, devotion to Rome above self. Sounds a lot like some later empires and nations. I'm curious if relative tolerance (cultural/religious) is common to many great empires/nations (for instance, Mughal Empire, Alexander the Great's Macedonian Empire, Athens during its Golden Age, British Empire, United States). Though there are clearly lots of exceptions (the Spanish Empire boomed during the Spanish Inquisition).