What is a republic government?

Republicanism is an old tradition of thought whose main idea was that the government should be a "mixed government". Later on, we started to linked it to the idea of democracy and elections. Another main conception of the Republicanism is the idea that the government should secure the Liberty of the citizens and the idea of the "civic citizen", who should be engaged in the administration of common affairs and the government. 
Although it was an already old idea, Polybius sintetized the notion of the mixed government for the Classical thinkers. He wanted to know why the Roman system was more stable than other. He used a tipology of the forms of government; however, unlike his predecesors, he said they were three good ones (monarchy, aristocracy and democracy) and three bad ones (tyranny, oligarchy and mob-rule) and that they were "simple forms" of government that were in an "anacyclosis" (degenerating cycle).
In other words, monarchy leads to tyranny, which will turn, eventually, into an aristocracy, then to oligarchy, which leads to democracy and, later, to a mob-rule or demagogy; at the end, the cycle starts again returning to the monarchy.
What Polybius proposed was tha the best form of government was that of the mixed government: one that was formed by elements of the three good constitutions. He though that Republican Rome expressed this principle: they had the monarchy in their Consuls, they had the aristocracy in their Senate and they had the democracy in their popular assemblies. Therefore, Rome was outside the degenerating cycle as long as it mantained the equilibrium between the forms of government. Another one of Polybius' ideas was the check and balances between each of the components of the mixed government. 
 
 
Later on, Montesquieu proposed the idea that, at the end, was used by the American Consitutionalists when drafting the United States Constitution: the idea of the three branches of government. It was, essentially, a turn in Polybius' idea: the government would be not divided between elements of the types of government, but between functional branches: the Executive, the Legislative and the Judicial. For him, separation of Public Powers wasn't absolute either: each branch participates in the activities of the others by process of check and balances (for example, when the Executive uses its veto power). The key objective of this separation is to secure individual liberty from authoritarian rule. 
 
Therefore, both classical sources of the republican government insist in the idea of a mixed government capable of mantaining equilibrium in the exercise of power. More than providing citizens the oportunity to elect their representatives, the first preocupation of the Republicanism was to secure the liberty of the citizens. For instance, the writers of the U.S. Constitution, originally, didn't want to stablish a democracy but a Republic which were able to resist the disruptive forces of the popular will.
Nonetheless, we can discuss a more recent view on Republicanism and democracy with Pettit (1993). For him, Republicanism views liberty as being protected from opression by laws and State norms; also, it thinks it's necesarry to elect public officials, therefore, it views the voting process as one in which the voters judge different options and choose based on the quality of each one of them. In a similar fashion, Republicanism believes that the process of government is a process of deliberation guided by the idea of rational discussion and the public good. 
Although is not the subject in question, is important to keep in mind that diferent thinkers have diferent ideas on liberty and on what the government should do to secure it.
In conclusion, the republican government is that which main objective is to secure Liberty for its citizens through laws and regulation (liberty understood as protection from opression of any kind). Since the ruling class is also dangerous for the freedom of the citizens, Republicanis thinks government should have an "equilibrium" between distinct forcers: being these either "forms of government" (like in Polybius) or "branches of government" (like in Montesquieu), in the manner that each branch is control by processes of check and balances by the other branches. 
References
Godoy, Arcaya (fall, 1996). "Antología política de Montestiqueu". In Estudios Públicos, 62, 337-406.
Sabine, George (1961). History of Political Theory (3ed). New York: Holt, Rinehart and Winston. 
Pettit, Philip (1993). "Republicanism and Liberalism". In Australian Journal of Political Science, 28(4), 162-189. DOI: 10.1080/00323269308402274
 

A republic form of government is when people elect leaders to represent them in the government. In the United States, the people elect leaders at the local, state, and national levels to represent them in the government. These leaders are supposed to listen to the concerns and wishes of the people they represent. Then, based on the will of the majority, the elected representatives should vote accordingly. The advantage of this form of government is that it is easier to get things done. If all of the citizens had to directly vote on every proposed idea, it would be a very difficult and cumbersome process. With this method of government, people elect leaders, and these leaders then should reflect the wishes of the majority of the people that they represent.