The city of Rome in the ancient world was called the eternal city, because it lasted for so long. In light of this, the achievements of Rome are vast and variegated depending on time period. In light of this, let me list a few achievements.
If you asked an ancient Roman in the days of Cicero about Rome's greatest achievement, he would say that the greatest achievement of Rome was in its ability to rule. For example, Cicero concedes that others are better at things like poetry, philosophy, and entertainment. But when it comes to statecraft no one does it better than the Romans. When we look at the historical data, we have to agree. Rome ruled a vast territory and they were able to do so, because of their political and managerial skills.
If we asked a early Christian in sixth and seventh centuries about the achievements of Rome, he would say that one of the greatest achievements of Rome was the leadership of Rome within Christendom. This bears some truth too, because Rome became place of the Pope and played a vital role in the growth of Christianity.
If we take a broader approach, we can mention many other achievements such as: Roman engineering (we still use Roman roads), construction of aqueducts (the longest one was 250 km), a Roman military ability. The list can go on, but these points should get you started.