For all practical purposes, the words "democracy" and "republic" are synonyms, and should be used as synonyms. That's demonstrated nicely in a fine new post by Steven Taylor, who focuses on explaining how James Madison used those words in the Federalist Papers. Taylor and I are both listening to democratic theorist Robert Dahl. As I've said before, the difference between the two is that one is from Greece and the other from Rome, but since no modern nation really bases anything they do on Roman or Athenian institutions, there's no point in maintaining some sort of separation between them.
The best practice, as far as I'm concerned, is to simply refer to different forms of democracy: direct democracy, representative democracy, majoritarian democracy, etc. Moreover, there's nothing inherently "pure" about one or another form -- none of them, that is, are "pure" democracy.
At any rate, I completely agree with Taylor: "In terms of contemporary usage, especially from a political science point of view, the formulation 'we have a republic, not a democracy' is almost nonsense."