What were some weaknesses and strengths of the Roman Empire?
This question likely refers to the whole history of Rome from 8th B.C.E to the 5th century C.E. despite the reference to the Empire that existed from the time of Augustus to the Barbarian invasions in the West and even longer in the eastern Roman (Byzantine) Empire. To this end, it is better to have more information, especially as each phase of Roman civilization (monarchy, republic, monarchy) transitioned from from the last, creating very gradual changes.
Some strengths of Rome were its military power. Not only was Rome's professional army unique at its height, their tactics and commanders were second to none. The various legions of the Roman army were loyal to Rome and not their commanders, so the fight for their civilization and way of life gave them a tenacious fighting determination. Close range weapons (gladius) worked in tandem with the shields and spears designed for long range disruption of enemy formation. Romans also learned from their enemies, be it the development of ships in the First Punic War or how to counter the threat of Hannibal's war elephants in the Second at Zama.
Romans were also skilled administrators, building the roads and communication tools that were necessary if they were to have large areas of land under the control of one people. Even though Rome largely expanded due to the threat of enemies all around their borders (which naturally persisted as the Empire expanded and during periods of shoddy Emperors), their systems were effective to the point that it would take more than a powerful enemy to bring them down. It is almost a boring answer, but Rome more destroyed itself through internal problems than it was outright conquered. Their enemies merely noticed and took advantage, as a system can only persist if the people are dedicated to it. Apathy is the real weakness in the fall of many an empire.
Other weaknesses included power too concentrated in the Senate and later the Emperor. While consolidation of power works with benevolent leaders who serve the public rather than their own ambition (Cincinnatus) it was rarely the case with the patrician class, what with the wealthy Roman senators serving their political ambitions rather than the good of Rome, the ability for few to climb the cursus honorum, and the flagrant violations of the rules designed to hold ambition in check (ie: no repeated holdings of offices; longer term limits) in the final generation of the Roman Republic. During the specific Empire period, there was also the choosing of Emperors by the Praetorian Guard, which caused short and often chaotic reigns, as these guards' favoritism would wax and wane, especially as they were the military presence within the city of Rome during this time. This made administration difficult and brought the notice of the enemies of Rome.