I'm sure there are a million citations of what a democracy is but I'll give you what I learned in college (even that can be construed as opinion)
A pure democracy is one ruled by the people where the public votes on every issue (referendums for everything).
A pure republic is one ruled by a dictator.
Since we have a republic with a democratic form of government, we vote for the "dictators" to make the voting decisions for us. This was applied because of the manner in which it wouldn't make sense for people in New York to vote for a new street light in Arizona. I hope that helps.
I have no problem with that definition. As currently used, there's no real difference between "Democracy" or "Republic." Both refer to a system of government where the people elect representatives within some sort of constitutional or legal framework.