The concept which flourished during the most glorious periods in Republcan Rome and that appeared on the Twelve Tables of the law, one of the first, though as yet imperfect, affirmations of the rights of man.
—Ernesto Teodora Moneta
There is no king but only law.
—German Historian describing Medieval Iceland
If The Alliance is the Five-Man Band of political entities, this one 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 country rather than separate states, 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 dictatures officially are republics too. Even those where the dictator manages to make his offspring his successor.