Rome was very successful because of its superior military authority. That is the main influential factor that cemented Roman power for nearly a thousand years. They had several advantages - rapid development of the latest technology and vast numbers of infantry along with a stable senate system and much wealth to fuel their ambitions. It had sewers to control waste, aqueducts for plumbing and paved roads for transport - which many other nations simply did not have.
Rome had a large assortment of choices when it came to battle. Their army usually consisted of Hastati, Principe's or Legionary Cohort in the later years - marvelous soldiers and very useful ones to say the least. Having said this, they had access to exceptional horses as well and it was this mixture of troops that was a deciding factor in conflicts. Rome used its sheer force of numbers to overwhelm most enemies and surround them or cave in on their flanks causing a route very quickly. Mind you, tactics were used, but not effectively.
The Romans took advantage of technological progressions and were only to happy to inflict deadly harm upon their foes with ballistae, catapults, onagers and numerous other siege equipment. The land they conquered poured much profit into the treasury which was distributed to the armed forces. This was able to give them the latest weapons, brand new armour, plenty of supplies and good morale. You can also look at the sheer amount of land they claimed through war - the empire covered Spain, France, Italy, Greece, Turkey, parts of the Middle-East, Egypt, and the Africa coastline.
Finally, the Romans brought the ideology of Christianity to the people in Europe - influencing the events that were to proceed in the future greatly, something that has changed the world today. They will be looked upon in 1000 years (during the Middle Ages) with their beliefs, attitudes, values, lifestyle and ingenuity being emulated.
In reality, the Roman empire has very little relevance or connection to modern day America. The time period is totally different - as is the cultures, way of life, economy and war. To compare the United States to Rome is ludicrous. What do they have in common? Absolutely nothing.
So it was very succesful because of its dominance in warfare and the stable structure of politics. Of course there is the decline of Rome, yet that is another story.
My own knowledge - The length is a bit long, but there are some very important aspects there to consider.