How did leaders react to revolutionary ideals in Europe after the French Revolution and the reign of Napoleon?

On the whole, they reacted with absolute horror. The crowned heads of Europe were strongly against the French Revolution from the start, seeing it as an attack on the established order of things. We should always remember when examining this period that most rulers believed themselves to be appointed by God. To topple the monarch was therefore seen, not just as rank treachery, but as a sacrilegious act against the divine order of things. A number of European states responded to what they saw as a dangerous development by declaring war on Revolutionary France. Not surprisingly, one of these was the Holy Roman Empire, ruled by Marie Antoinette's brothers Emperor Joseph II and, after him, Leopold II.
In the subsequent revolutionary wars, French forces swept across neighboring countries, taking large tracts of territory from their monarchist enemies—most notably the Spanish Netherlands. The counter-revolutionary forces were surprised by the ferocity and superior martial spirit of the French and eventually found themselves forced to sue for peace. The intervention on behalf of Louis XVI had been a failure.
If anything, Napoleon was perceived as an even greater danger to European monarchies. The Corsican general's famed military genius gave him an aura of invincibility in his early campaigns, and he was able to rack up a string of impressive victories against opposing forces, most notably when he crushed the Prussian army at Jena in 1809.
Napoleon proceeded to expand French territory considerably, mainly at the expense of Europe's various monarchies. The Holy Roman Empire—into whose ruling family Marie Antoinette had been born—was forcibly dissolved by Napoleon after his victory at the Battle of Austerlitz; it had stood for just over a thousand years, yet Napoleon swept it away and established the Confederation of the Rhine, a deliberately weak and fractured French satellite state.
Despite his enormous successes on the battlefield, Napoleon's grip on power was never completely secure. He had expanded French territory too far, too fast. The remaining European monarchies—Britain, Portugal, Sweden, Spain, Austria, and Russia, as well as a number of German states—formed themselves into the so-called Sixth Coalition which eventually defeated Napoleon at the Battle of Leipzig in 1814. The same powers formed the basis of the Seventh Coalition, which inflicted Napoleon's final defeat a year later at Waterloo.