The Democratic People's Republic of Korea, AKA North Korea. 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, but for different reasons. North Korea refuses to recognize the South as its own country, saying they're all race-traitors duped into servitude by the imperialist-pigs of the United States. South Korea similarly refuses to recognize North Korea as its own country, since doing so would be to admit that Kim Jong Un isn't a crazy man playing "dictator" with millions of peoples' lives.)
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.
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.
The Democratic Republic of the Congo, formerly Zaire. Highly atypical for a nation using such nomenclature, the Democratic Republic of the Congo has never practiced any form of communism, not even the In-Name-Only version used by modern China, though it's still got the usual lack of democracy. Its neighbor to the northwest, the Republic of the Congo, isn't much better off.
The Republic of the Congo (the one that's now the the Democratic Republic, not the other Republic of the Congo mentioned to above) 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.
Zaire was officially known as the Republic of Zaire, despite the fact that Mobutu ruled it as his own personal fiefdom, plundering every last cent he could from the country. Mobutu claimed that Zaire repudiated both communism and capitalism...in practice this meant it was his own personal brand of fascism.
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 inversion: 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.
Actual democratic republics in Europe (to the west of the Iron Curtain) inverted this. As illustrated by the former West Germany, such countries did not have "people's" or "democratic" in their official names.
France = French Republic (République française)
Italy = Italian Republic (Repubblica Italiana)
Austria = Republic of Austria (Republik Österreich)
Post-Estado Novo Portugal = Portuguese Republic (República Portuguesa)
Finland = Republic of Finland (Suomen tasavalta)
Switzerland = Swiss Confederation in English, German (Schweizerische Eidgenossenschaft) and French (Confédération suisse); Helvetic Confederation in Latin (Confoederatio Helvetica)
Yemen 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 "jumhūriya", from the Arabic jumhūr, meaning "public", with the -iya being a suffix that, in context,note It is also the suffix indicating the feminine possessive, but that applies in other circumstances. means "thing of (the) [thing it's suffixed to]" (thus jumhuriya is "thing of the people", i.e. a direct translation of the Latin res publica). "Jamahiriya" is a neologism based on pluralizing the "public" part (jumhūr) to mean "masses" (jamāhīr). 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. And defied by several actually authoritarian monarchies such as Saudi Arabia, Oman, Brunei, and Swaziland.
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 handful of regimes established before the closure of the Cold War, the 2014 pro-Russian insurgency in eastern Ukraine saw the establishment 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.
"Positive Christianity" sounds really uplifting until you know that it was a perversion of Christianity promoted by Nazi Germany.
The Constitution Party in the United States is a far-right theocratic party. The use of "Constitution" is ironic, since the US Constitution bans preferential treatment of any religion.
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.
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 '80s 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).
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. And within Canada, Quebec will be sometimes referred to as République Populaire du Québec for being more socialist than the rest of Canada and for their tendency to have to "do things their own way."
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...
One of the lines the anti-Ortega and anti-FSLN opposition in Nicaragua uses for their fuzzy non-Sandnista political concept note As oppostions go the Nicaraguan one is very heterogeneous and prone to infighting is "Nicaragua vuelve a ser Republica" (Nicaragua returns to being Republic) which first of all is extremely meaningless due to this trope, second of all a very dubious slogan given that the long form name of Nicaragua is "Republica de Nicaragua" and third a tad hypocritical on part of former Somoza supporters, who still play a role in the Nicaraguan right wing, as Somoza's regime can best be described as a Hereditary Republic.