When is a country Republican?

Since you used the lower case "republican", I assume you are referring to when a country is considered a republic.

The primary characteristic of a republic is representative government - citizens elect various representatives to some sort of central decision-making body, which then governs in the citizen's names. Citizenry does not have to be universal (or anywhere near universal), merely somewhat inclusive, to qualify as a sufficient base for political scientists to classify it as a republic, rather than an oligarchy (which is the closest form of government to a republic).

In modern usage, "republican" also tends to mean "democratic", in that the government is both a representative form, and has a fairly large citizenry, where membership in the citizenry is not excluded by reason of race or religion (though, for some reason, we allow societies which exclude women as citizens to be considered "democratic").