What the previous answerers didn't emphasize enough is that the real secret wasn't just the material itself, but above all *continuous maintenance* over the centuries. The Flavian Amphitheatre or the Pantheon didn't stay intact by magic: they were repaired, restored and repeatedly adapted by different generations. Romans also had a culture of "doing it right" - if a structure started to give way, they reinforced it instead of demolishing it. This means that most of what we see today contains materials and techniques ranging from the first century all the way to the Renaissance.
As for materials, the others are right about pozzolana, but there's an important detail: Roman concrete *improved over time*, especially when in contact with water. It's a particular chemical reaction, not just that it aged well like a normal brick wall might. But they didn't have the standardized quality we have today - every construction was a bit different, and many of them failed silently. How many Roman structures collapsed and nobody remembers them? The ones that survive are mostly those that happened to combine the right materials in the right proportion, plus the luck factor.
Another thing: Romans designed with enormous safety margins. Their arches, their pillars, were often *oversized* compared to what was necessary. Not out of fear, but because their building system was more "massive" than efficient in the modern sense. This meant that even when some parts started to deteriorate, the overall structure still held up for centuries.