Republic is a social policy in Civilization V. It is part of the Liberty tree, and initially available in Civilization V: Gods & Kings and later versions. In the base game, it was available instead after Collective Rule.
Adopting the Republican form of government has the power of inspiring common people to work harder. After all, they know that they are working towards their own good! As a result each city gains a Production bonus, especially when constructing Buildings.
As of the expansions, Republic is one of the starting policies of the tree, and it helps greatly with early development of new cities. The extra Production is available as soon as they're settled, and the first things they produce are usually Buildings. It could also indirectly help with building Wonders (which also qualify as Buildings), although the net result is less powerful than the Aristocracy bonus from Tradition.
A republic is a form of government in which people or the representatives of the people have ultimate power rather than the power being vested in a monarch or despot. The people or their representatives may elect a leader, but that leader's power derives from the consent of the people. Some classical cultures have had some form of republicanism, notably several Greek city-states and Rome (for part of its history), but for most of world history the monarchy was the most common form of government.
The republic reappeared on the world stage in the 18th century, when the United States divorced itself from the British monarchy and set itself up as a republic. As they gained independence from the Old World most of the new nations in Central and South America were formed as republics (Mexico toyed with a monarchy for a while but this didn't survive for long).
In the 20th century most of the European governments became republics after one or the other of the World Wars, as did many Asian nations. Many parts of Africa have since followed suit as they have shed their colonial masters. Today, most nations of the world follow (or at least profess to follow) some form of republicanism.