There are two separate but somewhat linked factors here: one is that, in the 19th century, countries (or more usefully for the purposes of distinguishing them, nation-states*) ruled by an Emperor called themselves Empires. The other is that states that ruled over other areas containing different nationalities (using the term in its 19th-century sense; think something vaguely similar to modern ideas of 'ethnicity') from the ruling state also called themselves empires. A third type (related to the second) was the colonial empire - essentially the main principle as the second, but overseas, ruled at a distance rather than up-close. The decline of both across the 20th century has led to the end of states calling themselves empires.

To use your specific examples: France moved from a Republic to an Empire in 1804 by crowning Napoleon Bonaparte (who was essentially an absolute ruler anyway by this point) Emperor of the French. France already had a colonial empire by this point, and had even managed to hold on to some of it through the Revolution and Revolutionary Wars, but the term 'French Empire' was now properly applied to metropolitan France, not its colonial possessions. In this way, we might say that the French Empire possessed an empire in addition to being an Empire. When Louis-Napoleon, Napoleon I's nephew, managed to seize power of the Second Republic, the transition to Imperial rule was accompanied by another change in name. France retained - and expanded - its colonial empire as a Republic after the fall of the Second Empire (although this was largely a different empire than it had been in Napoleon I's time, centred on Africa and southeast Asia rather than the Caribbean and the Americas), so the 'French empire' existed even after the 'French Empire' had been dissolved.

The other good example of this is Germany. Germany was inherently constructed as a nation-state, but it had an Emperor so one monarch (a Hohenzollern) could rule over the subordinate Kings who remained in parts of the new country. Thus it was an Empire, and founded on the almost-literal dead body of Napoleon III's French Empire. In the decades before WWI, Germany cultivated its own colonial empire. Germany's example is somewhat complicated by the presence of a large Polish population within the Empire; however, since this is a single minority nationality rather than many, generally we can see Germany as being an Empire of the first type and having an empire of the third. The German Empire ceased to be an Empire with the removal of the Kaiser, and lost its remaining imperial terminology in the latter half of the 1940s due to Allied decrees.

The classical example of the second type of Empire is Austria-Hungary. Vienna ruled a complex, multifaceted, and multi-ethnic realm stretching from Poland to the Balkans, incorporating South Slavs, Poles, Italians, Romanians, and of course the Hungarians and German-speaking Austrians themselves, who existed in an awkward and uncomfortable political union within the Dual Monarchy. Austria-Hungary had no colonies, and was by no means a nation-state; in many ways the Empire was whatever contiguous amalgamation of territory the Hapsburg family had managed to acquire and keep through into the end of the 19th century. There was no unifying national language, nor much real national sentiment, especially in non-German territories. It was an Empire because of the Emperor, but also because of its multi-ethnic nature. Once WWI shattered the Empire, it split into separate states (most of them Kingdoms).

Russia is a strange agglomeration of the first and second. While Imperial Russia was undeniably multi-ethnic, the state was primarily Russian, and conducted campaigns of Russification on non-Russian populations. The Emperor was Emperor of Russia and Tsar of all the Russias, plural. After the February Revolution removed the Tsar, it became the Russian Republic; once the Bolsheviks evicted the Provisional Government from Petrograd, it became the Russian Soviet Federative Socialist Republic, part of the Union of Soviet Socialist Republics.

Britain was a colonial empire. Indeed, there was no Imperial title until 1876, when concerns over Queen Victoria being socially outranked by her daughter, who was marrying the German Crown Prince and who would therefore one day be Empress. Even then, it only referred to India, resulting in the style of Queen-Empress or King-Emperor. Therefore, the 'British empire' technically referred only to colonial possessions such as India, and to Dominions such as Australia and Canada (although as the 20th century progressed, the term 'Commonwealth' for this second group of countries was more generally used), not to the islands of Great Britain and Ireland (however much of Ireland was actually under formal British rule at the date you're considering). Here, again, history intervened: the World Wars broke Britain's ability to sustain its colonial empire and Dominions and Imperial possessions gained independence. A general turn against the idea of Empires or empires in international politics led to the remaining pieces being renamed, and referred to, as Crown Possessions rather than colonies or imperial possessions.

*Nation-states are states whose political boundaries and authority are defined by their governance of a single nation. Nations, in a 19th and early 20th century context, are something like modern conceptions of 'ethnicity' - defined by history, language, and to some extent descent. The nation-state derives authority from its claim to be the expression of the nation, its government legitimate because it represented the will and interest of the national population, whether it is a monarchy or a democracy. While this is to some extent the default geopolitical unit of the modern world and thus familiar to us, it's useful to define our terms. Very few European states at the beginning of the 19th century claimed to be Nation-States, or articulated their political authority in that way, but by 1939 almost all did. Forms of democracy are one marker of a Nation-State, although they are by no means a compulsory or universal one; universal conscription is another, since citizens of a Nation-State can be said to have reciprocal responsibilities to the state as the expression of national will that the subjects of a monarch do not. Again, though, that's not a universal thing.

TL;DR: there were a few kinds of empires, but the things that made them empires stopped existing, so where the states still exist they're not called empires anymore.