It could probably be debated for a long time what the most important accomplishments of the Roman Empire were, and even what constitutes "accomplishments" (many of the people conquered by Rome did not consider that an "accomplishment"). But there are certain things that Roman culture definitely did accomplish which would not be exceeded until centuries later.
The first was the Empire's stability. The Roman Empire persisted for over 1000 years even at the most conservative measure---over 2000 at the broadest measure including the Eastern Roman Empire. That is astonishing, and essentially unmatched by any other civilization. Their military strength and their ability to maintain national security was exceptional. Based on the Greeks, they essentially invented modern military discipline and chain of command, and their military hegemony maintained for centuries a lasting peace across Europe called the Pax Romana.
Romans also founded many cities which went on to become global megacities today, such as London. Their language, Latin, went on to be the foundation of almost every language in Western Europe.
The Romans established infrastructure such as roads and aqueducts that lasted centuries---in fact, some of it still stands. They invented concrete and book binding, and made significant advances in medicine. They had a postal system and a social welfare system. Their system of law and government became the foundation of almost all modern legal systems.
Much of the world as we know it today would not exist were it not for the Roman Empire, which may be why it continues to fascinate so many people millennia later.