The letters SPQR stood for Senatus Populus Que Romanusa - the Senate and the People of Rome. After defeating Tarquin, the last Roman king, the people started a new type of government, a republic. A republic is a type of government whereby people elect officials to represent them in government.
As a publicity campaign, and as propaganda, the new senate had the letters SPQR chiseled, branded or stamped on everything that they controlled. Public buildings, new coins, and even park benches got the letters SPQR added to them.
The people of Rome were proud of their new government, and were reminded every time that they saw SPQR that they were a part of the Roman Republic, and no longer ruled by a king.