To keep things simple: democracy -> majority rule
republic -> majority rule + no inheritance of public offices (so no monarch)
Of course it can get more complicated than that - democracies can differ in terms of who can and cannot vote, freedom of press, how exactly the separation of powers is handled (or if the powers are separated at all), what method is used to distribute seats in the parliament, how much power is held by the president and how much is held by the prime minister etc.
By that definition US and UK are both (representative) democracies but of these two only the US is a republic and UK is a parliamentary ("constitutional" sounds pretty weird in this context given that there is no codified constitution) monarchy (while still being a democracy).