It would be wrong to call the Roman Republic a democratic system because they are quite different. A democratic system is a way of government by people in which citizens are expected to play an active role. The population is considered as a whole.
However, a republic system is a government in which citizens are considered as individuals and they elect representatives who can rule on their behalf.
When it comes to making laws in a democracy, the majority has almost unlimited power and there is no protection for the minority (in a strict democracy). On the opposite, in a republic, the laws are made according to the constraints of a constitution and by elected representatives (called Patricians). There is protection for the minority.
In addition, in a republic system the Patricians or Senators, were mostly aristocrats (wealthy class) and the highest position in this kind of government was held by two leaders (consuls) that ruled the Roman Republic. In a democracy that doesn't happen.
The Athenian democracy in Greece is an example of the democratic system and the Roman Republic is an example of the republic system.