A republican form of government is a type of government in which the citizens of a country have an active role in the affairs of the government, and the government is not headed by a hereditary ruler such as a king. This definition leaves a lot of room for wiggle room, as you can see; many governments around the world are considered to be run along the lines of a republican form of government, ranging from dictatorships to representative democracies. Many people think that the republican form of government is a particularly strong and effective form of government.
This type of government takes its name and inspiration from the Roman Republic, which actually involved the input of only a very small number of citizens. Many people in the West idealize Greek and Roman culture, so it is perhaps not surprising that the republican form of government has become so popular. Fans of this form of government argue that when well run, a republic truly represents the will of its people, and it can be easily changed if citizens desire changes.
There are lots of different types of republics. Some are governed by a constitution, for example, in which case they are known as constitutional republics; these governments are often headed by a President, and they may include branches of government which oversee legislation and judiciary matters. Others are collectives of numerous states, as in the case of the United States, and these are known as federal republics. Parliamentary republics have a separate head of state and a head of government who share powers with each other and with a parliament, while Islamic republics are run in accordance with Islamic law.
The strength of the republican form of government relies on the people involved. In some countries, citizens are extremely active in their governments, working to reform things they dislike, participating in elections, and holding officials accountable for their actions. In other nations, people are not as active, either out of a lack of will or understanding or because the officials in the government keep them in a state of fear or confusion. The “people's republics” in some Asian countries are one example of this sort of republican government, which is in theory run for and by the people but is actually run by a small and elite group.
There are lots of ways to set up and administer a republican form of government. The important thing to remember is that such a government is not necessarily a democracy. Democracy as a concept is often integrated into republican forms of government which try to be fair to their citizens, but it is also possible to see abuses of power and a ruling class in a republican form of government.