Republics are countries which are not monarchies.
The UK has a democratic setup and it fulfills all the requirements of a Republic but in theory it is still a Monarchy though they're just Ceremonial which I think they have kept only for nostalgia's sake.
So all democracies are not technically Republics(though the distinction is only superficial as in the case of the UK). And all Republics are also not democratic like People's Republic of China.
We, however, are a democratic republic.

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He that increaseth knowledge increaseth sorrow (Ecclesiastes 1:18)