“Republic” is from the Latin “res publica,” meaning “public thing.” It refers to what’s in the interest of the public, the common weal. Republics don’t have to be democratic, though they usually are in some way. All it truly means is that the state isn’t owned by nor run in the interests of a group narrower than the generality, at least in principle, as opposed, for instance, to tyrants and despots.
A reading of the entire address reveals Adams's breadth and depth of understanding of world views, of previous governments, and of the ideas essential to liberty for members of a free society--ideas which today's politicians either do not understand, or are determined to censor from public dialogue.