Alright, so I don't really have any issues with John Green, considering he does 10-15 minute youtube videos he isn't bad. But his coverage of the Holy Roman Empire is specifically awful. But anyways lets get to the question.

Holy

This was added by Emperor Frederick Barbarossa. He wanted to increase his influence in Italy and specifically over the Papacy; and thus wanted his Empire to be seen as the premier Catholic monarchy. It was a stylistic addition, think Venice's name "Most Serene Republic of Venice."

Roman

This refers back to Charlemagne being crowned Emperor of the Romans by the Pope. Charlemagne's Empire was arguably the greatest Empire in the west since the fall of Rome. Later the first emperor of the HRE, Otto I, took control of Italy and Germany and was thus crowned Emperor of the Romans much like Charlemagne was many years ago. The name just stuck after that, even if the Empire didn't control Rome or most of Italy.

Empire

It was an Empire. It could collect indirect taxes, tax trade, levy men and more money during war time. The Emperor was the highest law of the land, in both religious and administrative affairs. There was a parliament that could make laws (the Imperial Diet). The Emperor was the de-jure overlord of his vassals, and could thus intervene in disputes between them (though they many times didn't). While the vassals of the Empire could have an independent foreign policy, an could fight in outside wars, they were forbidden from making alliances against the current Emperor. That didn't stop many of them from doing so, but that's not all that different from a vassal plotting behind their sovereign's back.