In my simplified world, there are only four types of government: Monarchy, Oligarchy, Anarchy, Democracy and of course a Republic.
Within these types of government many different economic systems can exist. Economic system examples: fascism, capitalism, socialism and communism. And of course these can vary and be blended with each other.
The picture below is the structure of the United States government as conceived by the founders. But now the Senate is actually a democracy because of the 17th amendment.
A Republic is a mix of Monarchy, Oligarchy and Democracy. The braces and the arrow in the above picture show this.
Days of attention: the number of days that citizens get to participate per year: voting and lobbying.
Rule by ????: This is the number of people in charge.
In the Democracy each citizen participates when they vote. In a Republic the citizen can and should participate by lobbying his representatives and voting. Do you have a relationship with your representatives? If not you are operating at a half a day per year of involvement with your government. If the citizens to not participate then the Republic slides into just being a Democracy. And Democracy is Mob/Majority Rule.
Also there are clear boundaries between the different branches of government in a Republic. These boundaries must be preserved, else the nation will slip into some sort of tyranny. That could be either monarchy or oligarchy. Again, it is the citizen's responsibility to be involved to keep government branches within their boundaries. Citizens must participate to keep their government on track.
Center for Self Governance - Short Version
This is the presentation that I used to write the above article!