Originally the term "people's republic" was an aversion of this trope, at least in the way it is commonly understood. The first governments to use the name were anti-communist regimes created mostly by nationalists from various non-Russian peoples of the former Russian Empire immediately following the collapse of Russian authority in western and southern Russia as a result of World War I. The word "people's" in their names could also be translated as "national" (the Russian word literally means "nation," but in the sense of "an ethnic community", rather than "a sovereign territory.") Ironically, it was Soviet Russia that destroyed all of them. A few years later, for some reason, the term "people's republic" was resurrected by Mongolian communists as a name for their new state (which was pro-Soviet), and it remained associated with communist and left-wing regimes ever since.
Often the words "Justice" and "Freedom" are added to the titles of the political party and government agencies.
The Democratic People's Republic of Korea, AKA North Korea. It's probably one of the best modern Real Life examples of the trope, since it includes "people's" and "democratic" in the name. Considering that recently deceased Supreme Leader Kim Jong-il, son of the first leader Kim Il-sung, has passed power down to his son Kim Jong-un, the "republic" part is also rather dubious.
The DPRK's name is four lies for the price of one - it isn't democratic; it isn't for the people; it isn't a republic (seeing as the head of state is a dead man and the ruling family is a monarchy in all but name); and it doesn't include all of Korea, only the Northern half.note (It should be noted, though, that South Korea is also guilty of that last one, being the Republic of Korea, and that both sides still claim the whole peninsula as theirs.)
The People's Republic of China, which today manages to be both a hyper-modern society and a communist single-party state. In 1949, when the Communists were deliberating over names for the new Chinese nation, they considered "People's Democratic Republic of China" until Zhou Enlai pointed out it was redundant.
It should be noted that China is no longer communist by the original definition of the term (and policy-wise it is considerably less socialist than quite a few European democracies), and does afford its citizens some measure of freedom, if only on the most superficial level.
Another thing worth mentioning is that the whole concept of "people's democracy" meaning-wise is essentially just gibberish, as the word "democracy" itself indicates the power of the people. As long as the term "democracy" is a loanword in the target langugage it may not be that obvious but since the actual Chinese translation of the word carries the original Greek meaning, it's easy to see why Zhou did not want Red China to become, in the eyes of an average citizen, a "people's people-cratic commonwealth."
The former state of Democratic Kampuchea, today and previously known as Cambodia. It was an oppressive dictatorship controlled by the Khmer Rouge, whose ideology was based on a bizarre rural-primitivist perversion of communism (instead of aggressively building up heavy industry like proper communists, they did the precise opposite and forced people out of the cities to go farm the countryside). During the terror reign of Pol Pot, more than 20% of the country's population was wiped out by famine, disease, and organized genocide.
The Senate and Roman People (Senatus Populusque Romanus, or SPQR) retained its original name once it became The Empire, even though the Emperor had complete control over the senatus’ decrees.
Even before then, the "people" referred to were free-born property-owning males, and Senate membership was controlled by what amounted to institutionalized nepotism.
While decidedly undemocratic by modern standards, it was at least intended to be an improvement over the semi-mythologized Roman Kingdom. This same aversion to monarchy was the reason behind the empire not dissolving the senate, and is the reason behind the emperor claiming to be Just the First Citizen. And as the Roman Republic was modeled after the democratic practices of the Ancient Greeks, many of the same issues with the "people" could be found in the Greek democracies that inspired the Romans.
The Republic of the Congo was, as a bog-standard dictatorship, formerly known as the People's Republic of the Congo from 1970-1991.
Congo is no stranger to this; during the bloody, oppressive rule of Leopold II, it was known as the Congo Free State. The actual meaning of "Free State" (or "État Indépendant" in French) had nothing to do with real freedom, rather it designated that the entire region was Leopold II's personal property, separate from his role as King of the Belgians. The indigenous population were...also treated as his property.
The name "Union of Soviet Socialist Republics" counts, to an extent. The name suggests that the USSR was a federation of "soviets", locally and democratically elected councils, and the head of state was officially the chairman of the Presidium of the Supreme Soviet, a body elected by separate soviets to discharge their legislative business. In practice, however, the USSR was ruled entirely by the Communist Party, and the Supreme Soviet was just a rubber stamp for legislation, at both the federal and subnational level.
Soviet satellites also played this trope straight: East Germany was a "democratic republic" and Poland, Hungary, Bulgaria, and Albania all had "People's" in the name. Upon making the transition to democracy, the East German states joined West Germany and took its name, "the Federal Republic of Germany", and the other four all removed "People's" from their names.
Romania, which had been the "People's Republic of Romania" from 1947 to 1965 (at which point it changed its name to the "Socialist Republic of Romania") decided to top them all and just go by "Romania," plain and simple, when Ceauşescu and his regime came crashing down in 1989.
Hungary may be an aversion: in January 2012, it dropped the "Republic of" from its official name, and at the same time became less democratic by restricting freedom of the press and changing the electoral law to benefit the ruling party.
Poland requires a special mention because the name "Republic of Poland" has been used officially since the 16th century and was first used in 1358 (in one pledge of allegiance to the king) when Poland was still a hereditary monarchy.
Yemen, mentioned in the page quote, was until 1990 divided into North and South, not East and Westnote Although on a map it was hard to see why they weren't called East and West instead. South Yemen (formerly a British colony) was the "People's Democratic Republic of Yemen". Yes, a Communist dictatorship with Islamic flavoring.
The Great Socialist People's Libyan Arab Jamahiriya, or Libya as it was known under the rule of the late Muammar Gaddafi. The ordinary word for republic in Arabic is "jumhuriya"; "jamahiriya" is a neologism based on pluralizing the "public" part to mean "masses," according to the link. The preferred translation is the Libyan Arab Islamic State of the Masses. It's no wonder that the country was renamed to just "Libya" when the rebels took over.
During the Russian Civil War, anti-communist nationalist states added "People's" or "Democratic" to its name to distinguish themselves from the Bolshevik-backed "Soviet Socialist Republics", for example the Ukrainian and Crimean People's Republics, and the Democratic Republics of Armenia, Georgia and Azerbaijan.
The Union of Myanmar: the former military junta was known as the State Peace and Development Council.
Voltaire famously observed that the Holy Roman Empire was the pre-French Revolution equivalent of this trope. Instead of a totalitarian dictatorship claiming to be a democratic republic, it was a loose federation of autonomous cities and principalities, located more or less exclusively in Germany, claiming to be the actual, genuine Western Roman Empire (give or take three hundred years). "Holy", "Roman", and "Empire" were the great political buzzwords of The Middle Ages, and the HRE's claim to any of those was rather dubious.
While it always maintained some influence with the Catholic Church, several Holy Roman Emperors spent most of their reigns fighting the Pope for authority, and after The Renaissance and The Protestant Reformation they couldn't even agree what "holy" meant in the first place.
Charlemagne, the first emperor, gave Rome away to the Pope, and by The Renaissance the Empire was basically reduced to Germany, a region which the genuine article had for the most part never even owned (there are a couple of German towns near the French border that do have Roman ruins, but that's it).
In a complete inversion, the top 10 most democratic and free countries in the world are mostly monarchies.note For those who care, the top 10 most democratic countries as of 2011 are (in order) Norway, Iceland, Denmark, Sweden, New Zealand, Australia, Switzerland, Canada, Finland, and the Netherlands. All of these except Iceland, Switzerland, and Finland have hereditary monarchs as head of state.
Around the same part of the world as the above, Sri Lanka has the full name Democratic Socialist Republic of Sri Lanka, but according to That Other Wiki it actually is a democracy, with the "Democratic Socialist" part referring to the social democracy form of government.
An Argentine military coup that deposed President Juan Perón was named "Revolución Libertadora" (in Spanish, "Liberating revolution"). Besides being a military coup, unlike Perón—who was chosen by the people in free elections—the "Libertadora" banned the very mention of Perón's name. Yes, exactly as it sounds.
Rafael Correa, president of Ecuador, tries to impose control of the press by claiming that the freedom of the press is a state attribute. This control is named "Democratic control" of the press.
The Independent State of Croatia, which was actually a puppet state of Nazi Germany.
There's an obscure political party in the U.S. called the American Freedom Party. It supports a white supremacist ideology.
Venezuela's full name, Bolivarian Republic of Venezuela. Name taken after Hugo Chavez's "socialist" party started to take over and change a lot of things, usually while saying that they endorse democracy. While the "democracy" thing has been a thing of doubt and rumours in Chavez's 14 years presidency run, after his death, the government has taken a toll with a lot of repression, revolts around the country and heavy takes of corruption. The "Bolivarian" part in the name alluded to Simon Bolivar (Venezuelas's Liberator back in the 1800's) body of thinking being taken as a foundational philosophy, and to be fair many of the goverment policies were based on Bolivar's ideas... The Theme Park Version of them.
While the term "people's republic" was long believed to be a thing of the past, kept only by a hadful of regimes established before the closure of the Cold War, the 2014 pro-Russian insurgency in eastern Ukraine saw the establishent of the "Donetsk People's Republic" and "Lugansk People's Republic" which, while definitely not communistic in nature, are definitely portrayed as People's Republics of Tyranny by pretty much all Western media (standing in striking contrast to what Russia has to say about them).
Though "people's republic" is usually indicative of leftist dictatorships, some rightist political parties in countries such as Switzerland and Austria have "People's" in their name, in their cases the Swiss and Austrian People's Parties. Ditto the People's Party of Spain and the People's Party for Freedom and Democracy of The Netherlands.
The main conservative parties of Japan, Australia, and Denmark have "Liberal" in their names. Ditto the Liberal Party of British Columbia, which has no affiliation with the national Liberal Party of Canada. The name of Japan's "Liberal Democratic Party" can seem particularly jarring in the United States, where "Democratic Party" has long had a left-leaning connotation.
Sweden's main conservative party is known as the Moderate Party.
The term "progressive" is typically indicative of leftist politics, although one of Iceland's primary center-right parties is known as the Progressive Party.
Similar to the above, there is also a hardline conservative party in Norway called the Progress Party.
Real Life Allusions
In the United States (and sometimes other English-speaking countries as well), liberal leaning areas—especially college towns—that are surrounded by conservative areas will be given the "People's Republic" label as a way of conservatives mocking liberals. In turn, said liberal areas will often start calling themselves that:
Cambridge, Massachusetts uses the name "People's Republic of Cambridge" in jest, sometimes in official documents. It is far from the only Massachusetts example, as Amherst (note the silent 'h') is another wildly liberal victim/perpetrator (naturally, which one it is depends on who's saying it and how it's being used) of this trope.
The Maryland town of Takoma Park (in the DC suburbs) is often called the People's Republic of Takoma Park, for things like declaring themselves a nuclear-free zone in the Eighties.
Davis, California (home to a branch of the University of California) is yet another nuclear-free city, in spite of the nuclear reactor on the university campus; they also build toad tunnels so comrade toads aren't left out (as featured on The Daily Show). See the PRD page on the Davis Wiki. After the 2011 pepper spray incident, the People's Republic of Davis title has lost much of its irony.
The "People's Republic" label is even more often applied to the famously liberal home of the University of California's main campus, Berkeley.
In addition, many people from the east side of town consider themselves residents of a further subset, the People's Republic of East Davis, or PRED.
By extent, the entire state of California is labeled the "People's Republic of California" (or "Kalifornia", because apparently replacing the C with a K is close enough to writing it in Russian) in reference to its central government being largely Democratic and its social democratic policies.
Same applies to Boulder, Colorado, being a university town in a liberal county, surrounded by conservative areas on nearly all sides.
Bloomington, Indiana is a shining example of this as well, as most of the rest of Indiana is very Republican and conservative. Election maps show this quite well. (Monroe County, the area where Bloomington is located, in the south-central part of the state, is relatively blue compared to the surrounding red.)
Madison, Wisconsin (site of the University of Wisconsin and state capital), a VERY liberal city in an otherwise moderate swing state is often called the People's Republic of Madison, a term first coined by one of the state's governors.
An even more extreme example may be found in Texas, with the People's Republic of Austin (state capital and home of the University of Texas), a VERY liberal city located in the middle of a VERY conservative state.
And yet another good example is the Arizonan city of Tucson, otherwise known as the People's Republic of Tucson by the rest of the largely conservative state. Not only does it moonlight as a San Francisco styled "sanctuary city" for illegals, but it more than once threatened to secede from Arizona with the rest of Pima County to form its own state (though obviously it never followed through) in protest against some of Arizona's more conservative doctrines (i.e. SB 1070).
Vermont is one of the most liberal states in the Union, electing the only admitted socialist to Congress (Bernie Sanders)note although he has shied away from that label since the mid-90's, and no European-style Socialist would EVER consider the Senator among their ranks. An older biography of Sanders focusing on his time as mayor of Burlington in The Eighties is titled The People's Republic.
In the Australian city of Melbourne, a left leaning inner suburb Brunswick is sometimes referred to as the People's Republic of Brunswick
Though generally played for laughs, The People's Republic of Cork has expanded enough for T-shirts of Che Guevara to become relatively popular. The kicker is that Cork is merely a county in the south of Ireland (though it does have the second largest city in the country, discounting Belfast).
Which one ought to, because it's not in the Republic.
In Scotland, the People's Republic of Fife and the People's Republic of Motherwell. These areas probably have the best claim to the name out of any in the UK, seeing as they actually elected Communist MPs.
For periods of around four weeks, a portion of North Carolina becomes the People's Republic of Pineland, a dictatorship fighting against a rebellion that serves as a training exercise for Green Berets, whose main mission is to train indigenous forces. Their legal tender looks suspiciously like Monopoly money.
Canadians, when being compared to the (generally for rightish) American politics, will sometimes mention they live in the People's Republic of Canuckistan.
British politicians used to snark that only nations with no concept of law or justice felt the need to have a Ministry of Justice. Come the Blair administration, and what did Britain create...