What precisely do you mean by that? What would be the difference between that and a Democracy?

I also ought to point out something about Republicanism.
It actually has two meanings, Slarti is using one, the rest of you another.
The classical system of Republicanism, as originated in Rome, and then found later in the Italian City-States, is a system whereby all "citizens" (not necessarily everyone who actually lives there) are heavily involved in the political process, but decisions are not simply taken in a voting system. There are clear limits of power.
Because it was a different system to Monarchy, over time the word became also a word for non-Monarchical government.
The founders of America didn't want a Monarchy, but they also didn't want a Democracy, so they turned to the only other system really known, a Republic. Republics are actually designed for city-states, like Rome, Florence, etc, so they had to change stuff to make it fit.
So, in America, a Republic is now seen as a system where powers are separated and the government clearly limited in certain ways. However, in Europe, a Republic means a state that doesn't have a Monarch.

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... I am surprised at your insolence in writing to me at all. You know, as I know, that I bought this constituency... may God's curse light upon you and may it make your women as open and as free to the excise officers as your wives and daughters have always been to me while I have represented your scoundrel corporation.
I have the honour to be... your obliged humble servant, Anthony Henley
- MPs reply to constituent, mid 1700s