India is a democratic country,and so called republic of India, is this mean free public of India ?

yes it means that

India is called a republic because it's constitution is in force, i.e is working. We became a republic only on 26th Jan 1950 , the day the constitution was enforced and not on 15th Aug 1947. so, I hope you understand the difference between republic and democracy.

Disagree with Ashish. India is a republic because it has an elected head of state i.e. the president. Coming to your question, I find it difficult to comprehend as to what you mean. India is obviously free in the sense that the country is sovereign and independent and free from foreign rule. By extension, Indians are also free. But this freedom has nothing to do with India being a republic as you obviously understand. The Constitution of India also guarantees certain freedom to the citizens. But that is another area of discussion.

Republic of India The Republic of India commonly known as India, is a sovereign country in South Asia. It is the seventh-largest country by geographical area, the second most populous country, and the most populous liberal democracy in the world.