Great source for this is The War of Wars: The Great European Conflict by Robert Harvey, but better, more indepth books to read The Habsburgs: Embodying Empire by Andrew Wheatcroft, though you best get used to cousins marrying each other, and Friedrich Heer's Holy Roman Empire.

But the simple truth is, by this point, the Empire had ceased to exist as a military body or a political state. It was primarily a ceremonial title, and this was due to a gradual lessening of power - this was turned into terminal decline (in the authority of the HRE) by the Thirty Years War and the Ottoman Invasion into Hungary and, eventually, Austria.

Put simply, the HRE wasn't really conquered, it was abolished. Francis II was a genuinely kind family man, but a second rate administrator and ruler. He abolished the title in 1806, not long after the legendary Battle of Austerlitz, through fear that Napoleon might assume the title; the Confederation of the Rhine had a great deal of power (in the region) and there was little to stand in the way of Napoleon should he choose to assume the title.

In terms of why the Habsburg Empire failed to defeat Napoleon, that is less complicated, and in many ways the crux of the question. In brief, a combination of exemplary leadership (provided just as much by a semi-meritocratic army structure as by any Marechal), huge manpower reserves contributed by the Grand Armee and the Levee En Masse, a genuine and quite high esprit de corps and aggressive campaigning on the French side, and old fashioned tactics, poor recruiting doctrine and internal politics led to Austrian defeats not only in the Italian Alps but at the Battle of Hohenlinden, complete encirclement at the Ulm, and Austerlitz.

This paints the Austrian Army in a bad light, but it is unfair; it had some of the finest cavalry in Europe, and Archduke Charles was a skilled and competent commander who would go on to hold his own against Napoleon at Aspern-Essling and Wagram. They were simply taken aback by the reforms in the French Army, and forced to adapt too quickly.

The Holy Roman Empire's decline is a book unto itself, but I hope I've given you some ground to stand on.

Edit: Misspellings and incorrect sourcing - The War of Wars by Robert Harvey, not The War of All Wars.

The HRE's centralized structure had been almost entirely torn apart by the time Napoleon rolled in between its member "elector" princes and Austria. They were not in any position to call upon an army of similar number or skill to what Napoleon brought to bear, especially with constant infighting and intrigue between the various states.

The German states themselves were not in a terrible state, with many soldiers being hired out as professional mercenaries as far away as the American Revolution (on both sides). Prussia especially had thrived during this time and contributed to 40% of the Coalition force at the Battle of Waterloo.

You can argue that the short-lived German Confederation was a successor state to the HRE too.

Really stupid question, but I need to know. Didn't the Roman Empire cease to exist much earlier? Like year 100 or so?

Thank you all for a delightful discussion. I love History.

Follow-up question:

In some readings I've done regarding the Napoleonic Wars, they refer to the HRE as Austria prior to 1806. Is this a modern term of convenience (much like using the name Byzantine Empire) or would that have been a term used at the time, and if so, how early can the HRE be called simply Austria?

These answers are amazing! Thank you all for the input, but please don't go just yet!

Someone mentioned the Thirty Year's War as a point of decline for the HRE. Would it be fair to argue the empire was in an ascendant position prior to this?

What is Voltaire's great quote about the HRE you referenced?

Part of the reason why the HRE was so decentralized was because to become Emperor one had to be elected from the rulers of the various kingdoms, duchies, and other regions it possessed. So each new Emperor had to make more concessions to those regions autonomy, making each new Emperor less powerful.