The term republic has many different meanings, but today often refers to a representative democracy with an elected head of state, such as a president, serving for a limited term, in contrast to states with a hereditary monarch as a head of state, even if these states also are representative democracies with an elected or appointed head of government, such as a prime minister.
The Founding Fathers of the United States rarely praised and often criticised "democracy", which in their time tended to specifically mean "direct democracy" (many or all decision made by voting by all citizens), often without the protection of a constitution enshrining basic rights and separation of powers.
The modern concept of liberal democracy often includes a constitution, basic rights, and separation of powers.
A republic is sometimes argued to mean a "mixed" form of government where democracy is mixed with oligarchy/autocracy. This may take forms such as an "upper house" in the legislature that is not elected and/or a constitutional monarch having limited powers. Such institutions were once common, but have more recently often lost all real power, or have been replaced by elected institutions.