As others said, Rome fell from within. The Empire was too large for it to be able to sustain itself, there were multiple revolts, etc.
IIRC, the Roman Empire actually split in half, with the Eastern Roman Empire turning into the Byzantine Empire while the Western Roman Empire basically just turned into a bunch of marauding states. This happened over the course of centuries, though; it wasn't like it was this one big huge event where one day, all of a sudden, the Roman Empire is gone.
Thing you need to know is that Rome basically ruled all of Europe and bits of Africa and the Middle East at its height. You know how difficult it must be to ship supplies and soldiers to the far reaches of their borders to maintain the peace? Eventually, something had to give and with a weak military, Rome couldn't stop the collapse. I also recall Rome being sacked by barbarians a time or two, which likely devastated the morale of the troops and citizens of Rome.
But again, it was a gradual decline, over the course of centuries.
Am I correct? Roman history isn't my thing.
