If The Alliance is the Five-Man Band of political entities, The Republic will likely be The Hero (or just thinks it is), The Lancer, or The Big Guy. The Republic is the close relative of The Federation. If it is not part of The Alliance, it is usually friendly to it. It is almost always enemies with The Empire. Like The Alliance, it sometimes forms out of La Résistance. This government, like The Kingdom, is almost never evil. If it is, it's nearly always because of some military coup or a group of corrupt bureaucrats. Sometimes, The Republic ends up becoming The Empire because of this. The Republic is a far more tight-knit nation than The Federation, usually acting like one nation rather than separate countries, but shares most of its characteristics. If The Republic and The Federation are both in the same Verse, they may not always be on the friendliest of terms. It is often based in most ways on the modern United States, and generally some kind of humanitarian, democratic capitalist society. In terms of technology, The Republic is often the most highly advanced group in The Verse, possessing the most intensive research programs and best-equipped military (aside from The Empire). If it is pitted against The Empire, they will usually engage in a Lensman Arms Race, and will often be militarily similar. The Republic is sometimes more prone to open warfare with The Empire. If there is no Empire, The Federation and The Republic may act as The Empire for each other. The Trope Namer can be attributed as The Galactic Republic, usually referred to as just The Republic. However, in trope terms it's more like The Federation. Not to be confused with the People's Republic of Tyranny, or Plato's work of political philosophy. Keep in mind that republics in Real Life can differ much. There are democratic republics (what most people think of first), but Venice, Genoa, and the Netherlands were oligarchic republics, and many dictatorships officially are republics too, even those where the dictator manages to make his offspring his successor.
- Star Wars has the Galactic Republic, which stood for thousands of years before being corrupted from the inside and transformed into the Galactic Empire. Later materials in the franchise chronicle the transition from the Empire into the New Republic.
- Xandar from GuardiansOfTheGalaxy. Comic depiction aside, what little we see of it is a pristine, cosmopolitan semi-urban landscape (it was based off of Singapore's Gardens by the Bay). However, it's not all great: apparently the local (heavily militarized) law enforcement can send you to prison without a trial (none onscreen, at least). It is possible that the Corps was pushed to desperate measures, as they were dealing with a severe terrorist threat at the time of the movie.
- A Song of Ice and Fire has the Nine Free Cities, as well as the slave republics of Yunkai, Mereen, and Astapor. Generally speaking, these states give their citizens more freedoms than the Seven Kingdoms of Westeros do, but at the price of legalized slavery.
- Game of Thrones adapts the republics of the Free Cities and Slaver's Bay more or less faithfully.
- Pathfinder has a number of examples, but the most archetypal and iconic is the Andoran Republic. Founded by Taldoran and Chelish rebels who were tired of living under tyranny. The state they've built is a fairly clear stand-in for the USA, ruled by a People's Council (which includes congressmen and representatives of the various trade guilds), and a Supreme Elect (president). They're one of the most predominantly "good" nations in the setting.
- Crusader Kings 2 features merchant republics, including Venice, Genoa, Pisa, and any others you might care to invent (just declare a mayor to be the lord of a ducal sized domain). They provide fewer soldiers, but produce more tax revenue than regular ducal domains. There's also a Game of Thrones mod that allows you to play as the various republics of Essos.